See discussions, stats, and author profiles for this publication at: http://www.researchgate.net/publication/15401459 The mechanical properties of actin gels: Elastic modulus and filament motions ARTICLE in JOURNAL OF BIOLOGICAL CHEMISTRY · JANUARY 1995 Impact Factor: 4.57 · Source: PubMed CITATIONS DOWNLOADS VIEWS 108 344 200 8 AUTHORS, INCLUDING: Paul Janmey Søren Hvidt University of Pennsylvania Roskilde University 345 PUBLICATIONS 22,628 CITATIONS 9 PUBLICATIONS 512 CITATIONS SEE PROFILE SEE PROFILE Josef A Käs Thomas P Stossel University of Leipzig Partners HealthCare 177 PUBLICATIONS 5,633 CITATIONS 289 PUBLICATIONS 18,773 CITATIONS SEE PROFILE SEE PROFILE Available from: Josef A Käs Retrieved on: 15 September 2015 Fax:617-734-2248.greatly depends on filament length but does not require crossshaft (SFB 266). markedly diminish the measured shear moduli. and HL07680.’ and the shear stress rized above lend themselves t o interpretation by widely acexerted in the periphery of a white blood cell extending protru.harvard.edu. links. SGren Hvidt**. Berlin. theviscoelastic properties of gels formedby purified actin filaments have been measured by five different techniques and five different instruments using actin preparationspurified separately in fourdifferent laboratories. pp. varies by orders of magnitude. determined to be approximately10 pm. the Whitaker Foundation. at filament Boston MA 02115. Thesefindings confirm that relatively isotropic F-actin networks are sufficiently strong to stabilize cells. Physik Department. Dietmar LercheSS. pascal(s). July 12. Issue of December 23. 221 Longwood Ave. tional diffusion. 1994.cepted theories of polymer rheology (17. The costs of links thatwould prevent translational (reptating)movements. reported rheologic parametersaboutactinfilaments are widely divergentand confound interpretations concerning actin rheology’s contribution to cell shape and motility. Section 1734 solely to indicate this fact. Some investigators find that purified actin filaments that are presumed to be many micrometers in lengthat physiologic concentrations (2-10 mg/ ml) have shear elastic moduli of several thousandpascals. and 0-80336 Munchen 2. Dynamic light scattering datawere analyzedby a new methodto confirm that actinfilaments have nostable associations with each other andfluctuate in solution at a rate governed by the filament bending modulus or persistence length. Inc. Paris Ceder06 France. StosselSOn From the Wxperimental Medicine Division. QBGroupede Physico-chimie Thdoretique. Federal Republic of Germany. 151. 32503 . In most instancestheseinvestigations alsofind that higherstrains (>20%) and treatments that shorten actin filaments..sistent with this interpretation (19). dynamic light scattering. a phenomenon known as strain hardening. To address large discrepancies reported in the literature. Boston. A fundamental quantity. such as with actin filament-severing proteins. Erich SackmannllII. DK-4000 Roskilde. 14. Massachusetts 02115. Denmark. Anthony MaggsOO. I[ IIBiophysik. October 7.indicating that such perturbations account for low shear moduli and poor responsiveness to filament modifying treatments reported previously.E-mail: lengths close t o the transition betweenimpeded and free rotajanmey@fas. 0-1040 Federal Republic of Germany. DLS. and NATO. and depends very strongly on the length of the filaments and on the history of the sample prior to measurement. Josef KasSO. the Deutsche Forschungsgemein. Similar large forces are exerted by the cortex of locomoting keratocytes (2). Medizinische Fakultat (Charite) der Humboldt-Uniuersitat. Brigham and Women’s Hospital. It follows that F-actin icine Division. the absolute valuefor the rigidityof actin filament networks.Vol. The intracellular structure widely believed t o prevent cell collapse under such stress and to transmit high internal forces is a network of actin polymers (3-6). Shortening of actin filaments with gelsolin and mechanical perturbations reduce the shear modulus to low values identical to some reported in the literature. Roskilde University.: 617-278-0382.Tel. publication of this article were defrayed in part by the payment of page charges. depending on the observers. sions to engulf a yeast has been measured to be 1000 Pa (1). and that in a minimally perturbed viscoelastic gel. Boston Massachusetts 02115. depending on whether they were polymerized from ATP-conFluid flow along an arterialwall imparts a shear stresson an taining or ADP-containing subunits (16). actin filaments aresemiflexible chains that can manifest rotational diffusion which * This work was supported by National Institutes of Health grants elastic behaviorfromimpeded AR38910. 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 inas the strain increases(from 0 to crease in the elastic modulus 20%). The Mechanical Propertiesof Actin Gels ELASTIC MODULUS AND FILAMENT MOTIONS* (Received forpublication. Brigham and Women’s Hospital. 1994 Printed in U. Fluorescence microscopy confirmed that applying even small shear stresses to F-actin can orient and rupture thefilaments. In exhibiting these properties F-actinobeys rules generally accepted for gels in general (13) and qualitatively resembles the fibrin clot (12.C.for example. **Department of Chemistry.S. Harvard Medical School.Thomas P. This article must thereforebe hereby marked “advertisement” Recent high resolution videomicroscopic studies of F-actin directly demonstrating both entanglement and reptation conare in accordance with 18 U. rheologic studies of the viscoelastic properties of actin ought to have directrelevance to itsfunction in vivo(7). Therefore. The high elasticmoduli measured for F-actin in these experiments provided a base line todocument subtle changes in the mechanical properties of the polymers. But despite the agreement in principle about this function of actin filaments. Federal Republic of Germany ***Znstitut fur Zellbiologie. 51. 11 To whom correspondence should be addressed: Experimental Med. Departments of §Medicine and TlBiomedical and Biological Sciences. These measurements consistently showed that the elastic shear modulus of 2 mg/ml F-actin is on the orderof several hundred pascals. Manfred Schliwa***. THEJOURNAL OF B I O ~ I C CHEMISTRY AL 0 1994 by The American Society for Biochemistry and Molecular Biology. 269. HL19429.According to this view. 1994) Paul A. JanmeySOnll. 32503-32513.A. could stiffen a cell either by interpenetration or. through being immobilized by exogenous cross‘The abbreviations used are: Pa.S. 0-85748 Garching bei Miinchen. $Unstitut fur Medizinische Physik und Biophysik. a well characterized protein polymer gel. Technische Universitat Munchen. No.The findings summaadherent leukocyte of at least 10-100 Pa. long actin filaments are free to diffuse within a limit of constraints formed by their neighbors. The structures of individual actin filaments within gels very similar or identical to those studied rheometrically were also examinedby dynamic light scattering and fluorescence microscopy. as do shorter actin filaments bound together by exogenous cross-linking agents (8-12). and in revised form. Ludwig-Maximilians-Universitat. Ecole Supdrieure de Physique et de Chimie Zndustrielle de la ViEZe de Paris. 18). 16).3841). pH 7. In particular.Actin was polymerized by addition of 2 mM MgCl. Some investigators who found low moduli forpurified F-actin have reported higher modulifor F-actin inthe presence of cross-linking agents (25. Berkeley. as reported by these studies. Under these conditions the cells swellbut do not rupture. (28).In that case supposedlypure F-actin with a high measured shear modulus is possibly contaminated with cross-linking factors. The quantity K is the bending modulus of the filament. then models of secretion that invoke actin filament severing to facilitate vesicle movement (27) are not likely to be relevant.32504 Elasticity of F-actin and Filament Diffusion In contrast. Light scattering was measured a t 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. employedfive different rheometers. and the extract was centrifuged at 100. using cone and plate geometries and eithertitanium or stainless steel surfaces. Gelsolin was purified from human blood plasma by the method of Kurokawa et al. The investigators do not observe a large ratio of G' to G or strain hardening at small strains. andat moderate to high strains filament severing proteins. We summarize here the results for the scattering from individual filaments and then discuss the effect of hydrodynamic interactions between filaments on the scattering functions. When supplemented with 100 m~ KC1 (delivered from a 3 M KC1 stock) and warmed to 25 "C. Straindependent forced oscillatory measurements were also made using a Mettler-Toledo LS40DIN412 instrument and couette geometry (33). and. The cell pellet which now had a volume of approximately 7 ml was SUSpended in 15 ml of cold homogenizing buffer (0. there is no significant interaction between filaments. Light scattering data were analyzed by the method of Farge and Maggs (37) to determine the stiffness of the filaments. 2 mM EDTA. and thawing at 37 "C. pH 7. addition of gelsolin to purified F-actin or activation of gelsolin in extracts by Ca" very strongly lowered oreliminated the elastic modulus. Macrophage Extracts-Rabbit alveolar macrophages were obtained as described previously(29). t ) I '. pH 7. where B is an experimental base line. 5 ml of packed cells weresuspended on ice in 10 ml of TBS (10 mM Tris. and g(q.78. which may be as high as 10 mg/ml.n is the refractive index of the solution (1. 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 (1ps to 100 ms). and the average filament length is at least 10 times greater (19) as confirmed in Fig. 1) where q = 4m/A sin(W2).331.2 mM CaCl. Dynamic Light Scattering (DLS)-Light scattering (36) was performed on 6 m F-actin using a Brookhaven Instruments BI30ATN apparatus as described previously (8. when indicated.t) = exp(-q2G(t)) (Eq. Additional measurements weremade with a novel highly sensitive magnetically driven oscillator recently described (34. We compare directly the viscoelastic properties of purified F-actin with those of a cytoplasmic extract gel t o put the results in acellular perspective. mechanical perturbations. 0.and Munich.Unless actin was polymerized by addition of 2 m~ MgCl.34 M sucrose. 35). For long polymers arranged in an interpenetrating meshwork with mesh size 5 much less than thefilament length. however we can evaluate the integral approximately as follows: by changing the variable of the .321. 12. and 8 is the scattering angle. 0. DLS from dilute polymer solutions can be described by the intensity autocorrelation function Z(t. and the Ca2+-sensitive solation of these gels wasthe basis for the original isolation of cytoplasmic gelsolin (31). To try and resolve these discrepancies. Roskilde. (42). The function H ( q )is theFourier transform of the Oseen tensor (18) and takes account of the hydrodynamic interactions betweentwo parts of the same filament.4). The viscoelasticity of purified actin was measured directly after elution from the column. and 150 m~ KC1 to G-actin in solutions containing 2 mM Tris. 5 mM dithiothreitol. MATERIALSANDMETHODS Proteins-Actin was purified from rabbit skeletal muscle independently in four different laboratories in Boston.6. the shear modulus extrapolated from the data of these studies would be no more than 30 Pa. the mesh size is approximately 1pm.and Munich by the method of Spudich and Watt.2 mM dithiothreitol. H ( q ) = (-ln(Cqd))/4~q (Eq. as shown below. A is the wavelength of light (633 nm). Forced oscillatory measurements were done with both a Rheometrics RFS instrument and a Bohlin VOR rheometer. measurements of DLS are dominated by bending motions of the filaments (36. a quantity termed the persistence length. The integral (2) is analytically intractable. 2 m M ATP. we have performed a series of rheologic measurements on 10 different actin samples made from fivedifferent muscle preparations in four independent laboratories. or adenine nucleotides have little or no effect on the rheologic values (20-26). G(t) is a dynamic correlation function which expresses the propagation of density fluctuations along a semiflexible filament. other rheologic results that are inapproximate agreement with each other differ from those just described in nearly every aspect. or 150 mM KC1. conventionally set to 0. Moreover. The supernatant fraction was immediately frozen in liquid nitrogen in 800-pl aliquots. 0. further subjected t o gel filtration chromatography using Sephacryl S200 or a Superose 6B FPLC column. 3) where d is the diameter of the actin filament and C = 1. F-actin must be extraneously cross-linked. q ) = B + I g(q.For 6 p~ F-actin.0). or after freezing in liquid N. 10 mM Tris. a high elastic modulus was never observed ruling out a contribution of surface layers of drying or denatured protein a t the edge of the rheometer. The cells were broken by approximately 50 strokes in a Dounce homogenizeruntil few if any intact cells were visible in a microscope. such extracts form gels (30). Even taking intoaccount the intracellular actin concentration. the magnitude of the elastic strength of actin networks is more than a factor of 100 lower than the measurements cited above.5 m~ ATP. it would be impossible forit t o be responsible for maintaining theintegrity of the cell cortex. these maneuvers would not be expected to alter surface denaturation if it occurred. and 40 ml of water were added immediately prior to centrifugation at 1000 x g for 10 min at 4 "C. 100 mM NaCl. Furthermore. We find that reliable measurements are difficult or impossibleafter 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. An excellent recent summary ofhow these quantities can be derived is found in Gittes et al. 12). The Oseen tensor expresses the fact that movement of the filament causes a hydrodynamic flowwhich acts back on the filament even a t large distances.The implication of these results isthat in order to exert resistance to high shear forces. if actin rheology does not depend heavily on filament length (24). since the elastic modulus of protein gels scales with the square of the protein concentration (9. If an actin network has an elastic modulus of 1 Pa. Free oscillations were measured by torsion pendulums constructed and operated independently in laboratories in Boston.. after storage for 1day at 4 "C.Filament stiffness is most usefully characterized by the length overwhich correlations in the direction of the tangent are lost.261.000 x g for 40 min a t 4 "C. 10. Farge and Maggs (37) showed that the dynamic structure factor of a single actin filament was given by the following expression g(q. and supplemented our rheologic measurements with new methods for analysis of dynamic light scattering and fluorescence microscopyof single actin filaments. The principles of these instruments and theirapplication to actin gels are described elsewhere (9. t ) is the dynamic structure factor (43). 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. We conclude that the low values reported for the rigidity of F-actin are not due to actin purification methods or to differences in instruments used to make measurements but result from the mechanical disruption of actin filaments prior to or during the rheological determinations. and 150 r m KC1 to G-actin in buffer A. The integral should be cut off at a high wave vector corresponding to the inverse of the diameter of the filament. we used a dilute solution of rhodamine-phalloidin-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. This final result isindependent of the value of the parameter 1. we find 1000 ~ ' " ' 1 ' ' " 1 " " l " " l " " I " " l ' " ' ~ 100 (Eq. the slide and the cover glass were coated with G-actin. 6) where A is a constant we needed for further analysis. and the correlation time. 0 . A.7 cm). however. to stresses (force/area) applied parallel t o the surface of the sample. The images were recorded on videotape by a SVHS recorder (Panasonic) and analyzed on a Macintosh IIci.5 vol % mercaptoethanol were added after the solution was degassed for 1h.1 mg/ml glucose.exp(z41n(z~(t)/l%)/1)) z4 dzl where a ( t ) (4mTr7J~t)'~Cd.1 This is our fundamental result for the dynamic scattering due to a single filament or for a solution which is extremely dilute.7 pm. i. Michigan City. Fluorescent filaments were observed either with a confocal microscope (Bio-Rad MR6000) or 0. NIH). t ) )as a function of q2(t1(t))3'4 we can hope to deduce the elastic constant K. The pipette tip was cut to prevent filament breakage (diameter of the pipette after cuttingthe tip: 0. and 0. the buildup of G' during polymerization of a variety of different preparations of actin was measured in three different instruments. B .t)(l + A ( ( q o t ~ z ) / 4 ~ ) ~ ) (Eq. measurements similar to those shown in Fig. 20. Numerical work shows that a factor of 2 variation in t or K leads to a variation of less than 2% in this integral. a typical value found in our experiments is Z = 3.Image analysis was carried out with a modified version of the Image processing software (Wayne Rasband. Farge and Maggs continued numerically. only moderately successful. This is the main result of Farge and Maggs (37). a Zeiss Plan Neofluar 63xPh3 objective and a Zeiss HBOlOO light source. Expanding 2 in powers of 40 gives g. l . Such a scaling plot with our experimental data was. The integral should be cut off at an unknown lower wave vector which is independent of the scattering vector. but further analytical progress is also possible. A. and 44.5. Time course of increase in elastic modulus. IN). m 10 0 10 20 30 40 Time (min) 50 60 B 1 m a I Y Z 0.01 0 10 20 30 40 50 60 70 Time (min) FIG. P. with an inverted Zeiss microscope (Axiovert) equippedwith a filter set for rhodamine fluorescence. MTI.t) =gt. Similar measurements were made in a torsion pendulum at resonance frequencies between 1 and 10 rad/s and a maximal strain amplitude less than 2%. 0. m. To take into account this hydrodynamic screening we cut off the integral in Equation 2 at a lower.The symbols are as described forA.20 pg/ml glucose oxidase. CZosed symbols denote that the samples were gel-filtered. 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. constant value qo (which is only a function of the concentration but not t or q). To observe whether transfer of F-actin by pipetteting caused filament breakage or alignment.52. 4) 1. RESULTS Magnitudes of Elastic Moduli-The mechanical propertiesof polymer networks can be described quantitatively by measuring their resistance to deformation in shear. we chooseit in such a way that the integral in Equation 4 is almost independent oft and of K.(q. there are significant deviations from the theoretical result. In the following analysis of experimental results we shall use the following formula for the analysis of light scattering data. By plotting ln(g(q. Data fitted with the analytical form showexcellent scaling properties and can be used to measure the persistence length with confidence.22. The samples were put between a glass slide and a cover glass separated by about 100 pm and sealed by vacuum grease.4 mg/ml) by slowly sucking both solutions into a 1-ml pipette. and Z denote results reported in Refs. m. Perceptics. q... and 0 were obtained using actin prepared from three different acetonepowders and measured in an RFS rheometer at a frequency of 1rads and a maximal strain amplitude of 1 or 2%. at small angles however. The elastic resistance to such deformations is defined by the shear modulus G ( t ) (stressktrain. Knoxville. The selective labeling of filaments in an unlabeled F-actin solution permits the observation of the dynamics of a single filament. For documentation the Axiovert microscope was connected to a SIT-camera (SIT68.e.To do this we set it equal to the typical value of the logarithmic term in theintegrand for the values of the physical parameters that interest us. The long range nature of the hydrodynamic interactions is such that the system is very sensitive to the structureof the solution at large length scales. Thus we shall choose 1 = l ( t ) = ln((4~q/3~t)"Cd). TN). To prevent bleaching of the fluorescent dye. and N denotes data from Ref. 1:l) was mixed with semidilute solutions of non-labeled F-actin ( c = 0. The parameter Z(t ) is aweakly varying function of the rigidity of the filament and the time scales involved in the experiment. To impede adsorption of F-actin on glass. Fluorescence Microscopy-A very dilute solution (actin concentration c = 1x mg/ml) of rhodamine-phalloidin-labeledF-actin (molar ratio actirdphalloidin. 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.1.(q. respectively. supplemented with a fast frame grabber (Pixel Pipeline. The symbols 0 . 21. The symbols S.1-1. For large momentum transfers (large angles) the results do followthis theoretical law. 4 pg/ml catalase. and denote measurements made with a torsion pendulum in Boston using actin prepared in Boston with (El3) or without ( 0 modification by biotin orin Berkeley ( 0 . All samples contained 2 mg/ml actin and were polymerized within the rheometers by adding 2 mM MgCl. We thus evaluate the integral once and for all for this value of Z and find numerically the results 1 = 1. The measurements denoted by the symbols A. where strainis a quantitative .ElasticityFilament of Diffusion and F-actin 32505 A integration from q to z = q (t~U47rq)"where 1 is an arbitrary parameter. and kindly provided by David Drubin. t. To equilibrate the sample from strain and shear alignment we waited for 2 h until the sample was investigated. actin contains between 5 and 8 nM CapZ (46) and Measurements at 1 rads in an RFS instrument were made using a since some formsof modified actin monomers alsoact as caps at single preparation of gel filtered actin and adding various amounts of the barbed end (47). the values of G' for different actin samples reciprocal of these frequencies (0. The average value of these solutions of relatively short (800 nm. Fig. periments have been done. quencies. even though times greater than that reported in another study (22) (2 in their presumed filament lengths are much longer. and F-actin can also suspend small metal spheres which exert a shearing stress of strongly suggests that both the high shear moduli and the variability between samples are a resultof differences in aver.44)(points 3C) and(20) (Fig. These two implications are at odds with common observations of the materialproperties of F-actin. Mechanical Loss of Actin Gels and Their Disruption by Large the preparations which cannot be removed by gel filtration (46).5 to 4 pm. by varying the gelso1in:actin ratio changes the elastic shear modulus by more than afactor of 100. The value of 158 Pa is 1000 results arecloser to those of short actin filaments. The ments exhibit a much lowerG' that issomewhat more frequenaverage value of our measurements using multiple differently cy-dependent and.ElasticityFilament ofDiffusion F-actin and 32506 Effects of Filament Length-Fig. 4 demonstrates that F-actin can sustain free mechanical oscillations after transient displacement with an independent measurement of filament length.sistance to deformation. Atthe high Fig. A comparison with the measurements in studies ((21) (Fig. or purified from a different acetone powder. G'. 1B shows that theenormous difference in elastic moduli would cross at only slightly higher frequencies. (20). and there isno difference in results using the nearly independent of the frequency. overa large range of fresame actin preparation in different machines (see also Ref. the values of G' ing that oscillations or traveling waves cannot be sustained in drop by approximately 2 orders of magnitude. Effect of filament length on shear modulus of F-actin. frequency near 1 r a d s does not satisfy the most rudimentary 45).suggests that their identical experimental conditions. which remains stored in reproducibility of multiple measurements from the same an elastically deformed state. meanproduce an average filament length of 2 pm. 2b of Ref. . and is much larger (>50 times) than G over the After 60 min. / . whereas the the mechanical energy in the network. the low values of G' reported by some groups (20-22)could result from a 0 2000 40008000 6000 10000 relatively modest amount of filament breakage or contamination by as little as 0. 3B shows that sample is within a factor of 2 (32). t r' . These features imply that there light scattering show that greater than 95% of the actin has are few molecular motions with a time constant similar to the polymerized (91. 1 . 3 0 ) )on actin polymerized without or with a marked S. Fig. The gel-filtered actin samples (solid symbols) shows that F-actin networks with long filaments and a high tend to have somewhat higher shear moduli. P . which most of these exG' now nolonger depends on whether the actin was gel-filtered. / ~ . respectively. 2 shows that the shear modulus of a constant amount of F-actin depends very strongly on the average filament length ( L ) . but this is not storage modulus.Changing the average length of F-actin from 0.2. Second. by the moduli and the relative contribution of elastic and viscous restorage shear modulus. Because of 0 1 1 1 1 1 . 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. and concentrations lower than 34 p ~ at. a material with G > G' at a ments because of its ability to nucleate and cap filaments (8. which accurately determines the average length of fila.33 Pa such a material. are as described above. a clear interpretation of the rheology of pure F-actin requires Deformation-Fig.and out of phase components of the complex modulus. Only filaments long enough t o be immobilized by their neighbors contribute t o the elasticity of the sample. Since. Experimental conditions preparations. 20-22. to 0. Even at and now are very close to the values reported in Refs. the strong dependence of G' on filament length. such effects may be significant in many gelsolin to vary the average filament length. suggesting that G' and G' ported in ref. When fiveof the same actin samples as used for the data in definition of a gel (13).F-actin can maintain its own weight frozen. The variation among different samples is also much less. G is steeply increasing and nearly equal in magnitude to G'. Another aspect of actin rheology in which there is a large disparity of results is the frequency dependence of the shear measure of deformation) or. lA)on a similar sample using the same method as re. 9). lA are polymerized with a 1:740 molarratio of gelsolin to when subjectedto a transientmechanical displacement.important implications.greater than 10 Pa (48).end of the frequencies they report. Previous measmeasured by different groups largely disappears ifwe first urements (Fig. gel filtration. the inphase ments using eight different actin preparations and four differ. such a material isoverdamped Fig. 3A ent instruments. 1 . and measurements were begun 90 min after initiating polymerization. .21. This result when a tube containing it is inverted. . 1)under what were reported t o be 1:lOOO molar ratio of gelsolin. exhibit two interesting features. 1:300 gelso1in:actin)filasamples was 158 Pa with a range from 20 to 420 Pa.01-50 s) that can dissipate vary by more than afactor of ten from each other. as measured by G' and G .which G > G' at a frequency of 1r a d s (23). when pyrene-labeled actin fluorescence or entire range of measurements. Fig. even after FIG. and N in Fig. for oscillatorydeformations. values of G' reported in threerecent studies (20. In contrast. Fig. 1shows how G' increases during the polymerization of actin in eleven different measure. These discrepancies suggest that the age filament length. Since the average filament length cannot low shear moduli reported for F-actin networks in some studies be controlled using only purified actin and depends on the may not accurately represent theproperties of the unperturbed kinetics of polymerization and trace impuritiessuch as CapZ in material. ~ .3% of a capping protein. This result hastwo solin. 23) actually show a crossover at make filaments short using the filament severing protein gel. is not much larger than treated and measured samples is 100 times greater than the G . G ' .G' is very always the case.82 2 0. First. most importantly. 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. 4. on the basis of quantitative miscalculations. L4 1. If the ratio of G'/G had been as low as those " reported in some studies (20-221.0. no oscillations could have been induced.3. if wefirst broke the sample either to measurement. since the relatively high elastic moduli might conceivably be 0 affected by cross-linking between reactive thiols of actin. and compares them to the resultsof Newman et of the pendulum arm. moduli.1 1 Frequency (rad/s) 10 sinusoid.1 1 10 Frequency (rad/s) 100 D 7 0. Fig. 3. 4 were obtained using nearly 100% biotin iodoac-1 etamide-labeled actin in which the only surface-exposed cys0 2 4 6 8 10 teine of actin hasbeen protected(11). G'. and the lower curve repre. Furthermore. and moment of inertia. at strains greater than approximately 20% (11). The storage (triangles) and loss (circles) shear moduli were measured in an RFS8400 instrument at a strain amplitude of 2% and at the frequencies denoted for samples containing 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 concentration of F-actin reported in Fig.06 therefore impossible to reconcile with the dataof Fig.1 1 10 Frequency (rad/s) 0. G'= 73 Pa.01 0. the largevalues of G' previ- . Although the magnitudeof G' calculated from free 3 oscillations depends on measurements of sample volume. 21 ( C ) or Fig. (20) report very little straindependence One sample had been measured periodically by similar oscillations during the course of polymerization(upper curue. the data of Fig. G / G ' = 0. C and 1 D . the simple observation of oscil2 lations is independent of any possible calibration errors. 5A com3A. Free oscillations of biotin-labeled F-actin. The first measurement confirms the strain hardening had been measured by such a method every10 minfor one hour and rupture we previously reportedusing different instruprior to the measurement shown. and ratio G'/G of the two samples are inclose agreement with each other and with the data of Fig. Newman et al. In both cases. Oscillatory measurements were made in a torsion pendulum using two similar reports differ radically is thestrain dependence of elastic samples of biotin-labeled F-actin 65 min after initiatingpolymerization. followed by an abrupt decrease attributed to network rupture a low mechanical loss near thatpredicted from the dataof Fig. and assumes that the apparatus itself does not damp the elasticresponse. the sample al.ments andis in contrastt o the resultsof Newman et al. A third aspect by which our results and thoseof some other FIG.1 100 C w u 0. whereas we have previously reported a strong increase in G' a t small strains other sample had been left undisturbed prior to this measurement.01 0.and H in Fig. In case the of the uppercurve. Frequency dependence of dynamic shear moduli.m ElasticityFilament of Diffusion F-actin and 32507 A B 1:500 gelso1in:actin 10 - loo h k v FIG.32). and previous studies using larger amounts of 4 gelsolin confirm that such materialsdo not support free oscillations (32).However. 3 A . Very similar results have Time (s) also been obtained using native F-actin (9. and the of G' over a range of strains from 1 to 200%. 0. (20) for sents a sample that hadbeen left undisturbed for 60 min prior strains below 30%. 4 of Ref. (20). 3 of Ref.1 1 Frequency (rad/s) 0.01 0. 20 ( D l . and = 5. Two samples of F-actin were polymerized between the pares the relative value of G' measured by oscillations over a plates of a torsion pendulum and momentarily deformed to a range of maximal strain amplitude takenon the same sample strain of approximately 2% by a pulse of air directed at the end at two times. and the frequency. the measured oscillations are by large oscillatory strains or by constant shearing to several well fit by a theoretical curve for an exponentially damped hundred percent in one direction. A. we also obtained data using a different rheometer fimbrin.4 Pa and are compared to the similar data (0)obtained from Fig. the shear modulus was measured at 1 r a d s in an RFS rheometer by oscillatory measurements at increasing strain amplitudes actin in the absence and presence of a 1500 molar ratio of gelsolin. stress was measured a t increasing (A) ent with an absence of permanent cross-links or other irreversstrains every 3 s during shear deformation a t a rate of 0. 2 Stress Relaxation of Actin Gels-An independent measurement of the viscoelasticity of F-actin samples was obtained 0 I 1 from stress relaxation experiments in which the sample is first rapidly deformed to a constant shear strain of 10% and the 10 100 resulting stress is measured as a function of time that the Strain (YO) sample is held in the deformed state. geometry. The composition of cell cause it isnot observed if the filaments are first made short by extract gels is complex. 3. the composition of proteins associated with the easily sedimentable component of gelled extracts has been analyzed same stresses (data not shown). containing not only actin filaments but gelsolin or when materials that are more resistant to rupture. and depending on the av. &I shows that such an extract strains. 8 shows the to increasing extents (Fig. Macrophage Extract Gels-If actin filaments dominate the ously observed a t small deformations are lost. 7 for 1. The ratio of stress to FIG. to the extent that such gels qualitastress when a sample was strained at a slow shear rate (0.in several studies. such as fibrin gels. develops a shear modulus near 1000 Pa after several hours. and other constituents. We confirmed viscoelasticity of the cell. equivalent to G' at a frequency of 1rads. followed by an abrupt loss of mechanical resistance. the network does not rapidly repair itself. small area nearthe axis of rotation at the bob. Again. 8B shows that the cytoplasmic extract gel exhibits an although if left for several minutes. consist50% which was 1. The formation of actomyosin rigor equipped with a bob and couette geometry in which nearly all complexes as ATP is depleted may account for the very high of the mechanical resistance is provided by surfaces far from elastic modulus observed at later times. tubulin. 30 p~ cytochalasin B.increase in G' at increasing strains (strain hardening)and an erage filament length.6. the macrophage cytosol formed by centrifugation at 4 "C is slowly stresdstrain curve shows significant strain hardening at small warmed and forms a gel. a-actinin. isshown in Fig. The initial resisting stress is much lower in the gelranging from 1 to 50% (A). and is mostly actin and substoichiometric duced by sample drying at the edges of the sample in conelplate amounts of actin cross-linking proteins such as ABP. completely prevents gela- . 16). In close agreement to the re. then the elastic moduli measured for that F-actin networks undergo irreversible damage when sub. To verify that strain hardening was not an artifact intro. similar to our previous results increase and abrupt decrease is characteristic of F-actin be. 6 aresimilar to previous measurements observed in torsion pendulums and 4 in Rheometrics or Bohlin instruments employing either parallel plate or cone and plate geometries.strain. (20). 5B). where the shear stresses are negligible compared to surfaces at larger radii. a value roughly original position and the measurement was immediately repeated (A). the increase in G' at small strains. or vimentin networks are subjected to the However. Fig. and we obtain data very similar to those of Newman etal.5.increase in G' when a low ionic strength extract containing 30% sults of forced oscillations.02/s up to a ible changes in filament structure. once broken. Dependence of shear modulus on the degree of defor. are very close to the corresponding measurements shown in Fig. which is limited to a very which inhibits actin polymerization. the edge of the sample exposed to air. A and B.7 mg/ml mation.obtained with purified F-actin (9. The shear modulus is shown for measurements at increasing maximal strain amplitudes at a frequency of 1 r a d s using a MettlerToledo rheometer. the shearmodulus G(t). and the abruptdecrease at strains >lo% shown in Fig. The initial abrupt ruptureabove 7% strain.The rheometer plates were then returnedto their moduli of both samples at times less than 1 s. Under these conditions. and in both samples the resisting shown relative to the shearmodulus measured at themaximal strain of stress slowly relaxes after several hours nearly to zero. Fig.purified actin samples should be similar in magnitude to that of jected to large strains by a direct measurement of resisting a cytoplasmic extract gel. and myosin (49-51).02/s) tively resemble the intracellular actin gel. also actin binding proteins. B . and to our previous results. some recovery is observed.Additional measurements (A) were then made at decreasing strain amplitudes from 50 to 1%. Strain dependence of F-actin measured in couette geometry. The data are solin-containing sample.32508 Elasticity of F-actin and Filament Diffusion A 25 C'' 20 - 15 - 10 - I 50 4 40 5 k 30 v u i 20 10 0 0 1 B 10 Strain (Yo) 100 10 20 30 Strain (%) 40 50 FIG. 4 of Ref 20. The magnitudes of the shear maximal strain of 10. Fig. whereas 90 p~ phalloidin.1 10 1 1000 100 300 150 250 200 Time (min) 1 2 B Time ( s ) B lo 100 50 R -1 t\ j 5001 actin:gelsolin 1 3000 1 F h v w 0 0 0. Fig.the datawas restored. 9C. functions of light scattered from 6 p~ F-actin measured at Directly after pipetteting the sample onto the cover slide the angles from 30" to 150". at longer times and smaller angles.ElasticityFilament of Diffusion and F-actin 32509 A A 1000 h k h v v w k 100 c3 10 1 0 0. F-actin samples (1. on the order of a micrometer. there are large deviations due to hydrodynamic interactions between filaments in the solution. and the ment of filaments into bundles. with an error of approxiwavelength of visible light. Even the small fraction of labeled analyzed the results using Equations 5 and 6. a stabilizer of actin polymers.visualized by fluorescence microscopy. Oscillatory measurements were made in a torsion pendulum at strains of less than 1%after warming the extract toroom temperature (A) or at increasing strains 4 h after the gel had formed ( B ) . Stress relaxation of F-actin with and without gelsolin. q2(tl(t))3"and adjusted the coeficient A so that the scaling of Dynamic Light Scattering. filament-filament cross-links or lateral align(181. scattering angles. dynamic light scattering should be domi. interactions between different parts of the same filament. (39). The shear modulus (ratio of stress to strain) is shown during the time that the stress relaxes in the samples. but Despite the superficial resemblance of the filament intersecnot for effects between filaments. We have per. these entanglements are not static cross- . Piekenbrock et al.(42. 1OB and the network needs 2-3 h to relax into its shown in Fig. This value is in approximate agreement with Schmidt et al. This analysis accounts for hydrodynamic entangled.7 mg/ml) without (A) or with ( B )a 1:500 molar ratio of gelsolin were polymerized for 1h and then strainedin less than 0. Therefore. From the slope of the curve and average distance between filaments (the mesh size of the net. 52-54). and Piekenbrock measurements of thisquantity by fluorescencemicroscopy and Sackmann (401. 9A shows the raw normalized intensity autocorrelation trates the entangled state of an equilibrated F-actin solution. The superposition of the curves confirms that theonly filaments are so long that diffusion is severely restricted as interactions between filaments are purely hydrodynamic and predicted by theories of deGennes (17) and Doi and Edwards not. 3 4 Strain (%) 5 6 FIG. For short times and large tions to cross-links. however as noted in thetheoretical discussion. replot the experimental data to find a reliable value for the accelerates the initial increase in G (data not shown). nated by internal motions of individual filaments due to flexFluorescence Microscopy of Intertwined Actin Filamentsing. 1OA shows a typical formed DLS experiments at a variety of scattering angles and solution of actin filaments. we obtain the curves shown in Fig. The results are shown in Fig.filaments are still able todiffuse by reptation was directly sity autocorrelation of dynamic light scattering.When these data are replotted as a solution of actin filaments exhibits a highly aligned state as function of the scaling variable q2(tl(t))3/4. we plotted ln(l(t))/(l+ A (tl(t))1'4)as a function of gels is dominated by the actin filaments within them. entropic state. even when the mass concentration of actin is low. These elastic constant of an actin filament. 9B. the scaling of the data is good. Shear modulus of macrophage extract gel. and longer than the length of the actin filament of 16 pm. as recognizedby mately 30%.1 1 1000 10 Time ( s ) 100 FIG. All of ing Stiffness-Taken together the viscoelastic properties of in.from Equation 5 we can deduce a value of the persistence work) is large. and therefore the persistence length or equivalently the The formation of an interpenetrating meshwork in which actin bending energy of the filament can be obtained from the inten. for example. Determination of Filament Bend. filaments (ratio of labeled to unlabeled filaments 1:2000)illusFig.05 s to 10%. We must therefore tion. (36).the curves now superimpose over the entire rangeof times and tact F-actin networks suggest that they are elastic because the angles. In accordance with Equaresults confirm that the viscoelasticity of macrophage extract tions 5 and 6.8.7. The filament slides a distance which is comparable to its own length through the entangled network.Elasticity Filament ofDiffusion F-actin and 32510 1 - 0.6 .5 nM. entanglednetwork. The uncut FK:. Ihodamine-phalloidtn-laactions within ( B )and between ( C )filaments according to Equations5 beled actin filaments embeddedin an unlabeled F-actin solution of 2. The data inA are FIG. 11. as shown in Fig. 0. rhodamine-phalloidin-lalinks.0 ments diffuse through the entangled mesh of surrounding mg/ml concentration directly after the solution is pipetted onto the cover slide. 2.9. The filamentsform a non-cross-linked. The during pipetteting. we pipetted a very dilute F-actin solution (2.8 0.Experimental conditions are described in the text. reptation motion of a 7-pm long rhodamine-phalloidin-labeled filament through a matrixof unlabeled filaments of mesh size 0.t~1)/4q)"~) '~ /1013(SI units) FIG. would have a shear modulus above 70 Pa.0 mg/ml.5 h of relaxation time.2 n - 0 1 q' 2 3 4 ( t / ~ ( t ) ) ~(l+A((q. The filaments are in a highly aligned state due to flow F-actin by a snakelikesliding motion called reptation (19).Normalized intensity autocorrelation functions were obtained a t angles of 30" to 150' from 6 g~ F-actin (A). B .beled actin filaments embedded inan unlabeled F-actin solution of 2. R m e sequence of the reptation motion of a i. because the filaments can still diffuse.10. This finding excludes the existence of cross-links between the filaments.0 and 6 a s described in the text. Breakage of Actin Filaments by Shear Flow-The conformation of actin filamentsis drastically affected by the bulk flow of an F-actin sample caused by macroscopic mechanical perturbations. Fluorescence microscopy of single actin filaments in replotted against reduced variables to account for hydrodynamic inter-an interpenetrating F-actin solution.4 - 0. A . F-actin of this average filament length. equivalent to a 1:1850 molar ratio of gelsolin. To quantify the degree of filament breakage we measured the length distribution by video .l-pm long actin filament embedded in a solutionof actin filamentsof concentration c = 2.11. pipette tip has a minimal diameter of -1 mm and the cut pipette tip diameter is -7 mm. and how easily breakage of actin filaments takes place. Individual fila.1 pm is shown in Fig. To illustrate this effect. 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. stabilized by rhodaminephalloidin) with an Eppendorfpipette with two different diameters.Dynamic light scattering. because these would limit motion of the filament to lessthan themesh size of the network. in thefirst case we cut thetop of the pipette tip and inthe second case we used a normal 1-ml Eppendorf tip. Diffusion of an actin filament withinan F-actin meshwork. mg/ml concentration after 2. and several types of forced oscillators. 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 tipas shown in Fig.ElasticityFilament ofDiffusion F-actin and 32511 FIG.In this regime the network relaxes from strain by the reptation motion of single filaments. heated to 36"C for 2 h.proteins canbe reproduced by our experimental methods if the ratories and measured in various instruments we have repeat. The discrepant results in the literature appear rather tobe due to thepresence or absence of irrecoverable changes in the F-actin samples thatcan occur when the sample is placed in a device for making measurements or when a n initial large deformation is imposed on the sampleprior to measurements. At higher concentrations. The low frequency region is characterized by the transition to fluid behavior indicated by a drop of G'(o). For these reasons and since the actin used in our previous studies has been extensively characterized in many other assays. . The measurements were started after annealinga t 20 "C for 6 h. In the case of the pipette with thenarrowdiameter =50% filaments breakageoccurs as indicated by the change in the length distribution. large G ' / G ratios. Rheology of F-actin at very low concentration. In particular.5 nM rhodamine-phalloidin-labeled F-actin solution pipetted with a Eppendorf-pipette of 0. 11. and large effects of actin fragmenting maneuvers on breakage by shear forces. 44. as reported in some studies (21. Effect of filament breakage. and the minimal dependence of these parameterson the degree DISCUSSION of deformation of actin gels or the actions of filament severing Using actin independently prepared in several different labo.a p 0.01 10. The plateauregime extending from 10"' rads < o < 10' r a d s resembles rubber-like elastic behavior. strain hardening. The sample waspolymerized a t 20 "C for 4 h. 57).' Io-2 1oo Frequency (rad/?.) FIG. 12. thevery low elastic moduli reported by some groups. failure topurify and maintain native actin or other unspecified artifacts are eitherextremely prevalent orunlikely to account for the differences between our findings and those of other reports concerning the rheology of Factin. Fig.7-cm tip diameter (left side) or with an Eppendorf pipette of 1-mm tipdiameter (rightside).1 b 17 19 I : . The observed frequency dependence is in good agreement with the theoretical predictions (18) and was recently confirmed by microrheological measurements (56). where shear forces between filaments also contribute. However. Due to the very dilute F-actin solutionfew ifany entanglementsoccur and the breakage is caused by friction forces between the polymer and the solvent. The flow of F-actin and of microtu- . above the so-called semi-dilute transition. l i i <I I/ I 2 IS filament length (pm) 7 microscopy. Similar rheologic properties were observed in an actin-rich cellular extract. contamination by cross-linking proteins.05 mg/ml (55) filarnenUfilament contacts would greatly increase the forces applied to each filament and therefore increase theprobability that they break. The measurement wasperformed over five frequency decades and three regimes are clearly visible. I 7 h 7 x 9 1 0 filament length (prn) -1 - 0. a recentlyconstructed 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 semiflexible filaments.13. Frequency dependence of the storage modulusG' of an F-actin solution a t monomer concentration c= 300 pg/ml.gels are firstbroken or subjected to high flow rates. . Other findings corroborate the susceptibility of F-actin to edly documented high elasticmoduli. Operating below the accurate range of force transducers can be a significant problem especially for samples with low shear moduli and atlow strains and frequencies. as well as a stress relaxation measurement. and the effect should be larger for an entangled F-actin solution.Conventional rheometers arenot sufficiently sensitive to measure such relaxation under these dilute conditions in oscillatory measurements.Errorsin calibration are similarlyruled out since consistent results areobtained using freeoscillations in torsion pendulums of different design. Indeed our instruments would be incapable of measuring accurately materials with shear moduli of 1Pa a t < Hz at strain amplitudes of a few percent. and cooled down again to 20 "C. This experiment shows the lower limit of filament breakage. In this regime the dynamic elasticity is determined by the internal dynamics of the actin filaments. reducing the elastic moduli.12. The upper panel shows fluorescence micrographs of a 2. At higher frequencies G' again increases. Reptational Diffusion of F-actin duringOscillatory Shear-If the actinmeshwork is composed of individual filaments whose diffusion is retarded but not prevented. then lowering the filament concentration and the rate of deformation shouldreveal a viscoelastic relaxation duet o the centerof mass diffusion of the long polymers through the meshwork by the reptation process shown in Fig. which for such filaments ison the order of 0. which are therefore normally measured by creep experiments under constant stresses. 13 shows the frequency dependent measurement of the storage modulusG' of an F-actin solution of -7 PM concentration. The lowerpanel compares the resulting length distributions. Funatsu. 31113120 36. Chem.. 62.397430 4..61. Y. Acad. subsequent measurements even at low shear strains do not measure the same material as was present initially.. Methods 22.. (1993)J. E n . D. 62).. G. Chem. and Meld.. (1993)p. G. J. I.. S. Casella.. F.. T. the average filament length (64) as well as the mesh size (65) are on the order of several hundred nm.D. K.58).. 68. D. H. A transient overshoot in viscosity occurs when a rheometer is first turned on to high shear rates. The effects of shear flow on the structureof F-actin are most clearly seen in the fluorescence micrographs shown in Figs. Janmey. 21. H. D.1471-1489 41.801409 27.8988-8993 31. deGennes. 8218-8227 10. Janmey. Reu. W.. Doi. P. S. J. Peetermans. Newman.. H.Elsevier. American Institute of Physics. 282-303. J..(1993)Polymer Gels and Networks 1.. Chem. and the formation of anisotropic structures in actin filament solutions can result from shear flow (59.Miiller. A. 139-45 50. (1980)J . W. (1971)J. Cell B i d .205-214 26. Biophys.Hvidt.Cell Biol.. ed) pp.. and Pollard. Stossel. J. (1993)Biophys. Cell Biol.Polymersand Gels (Burchard. (1986) J. 0. Both the strains and strain-ratesof cytoplasm in a living cell (66. S. Stossel. N . 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Report "Mechanical Properties of Actin Gels - Elastic Modulus and Filament Motions"