Microrheometry of semiflexible actin networks through enforced single-filament reptation: Frictional coupling and heterogeneities in entangled networks

M. A. Dichtl† and E. Sackmann

Lehrstuhl fu¨r Biophysik E22, Technische Universita¨t Mu¨ nchen, James-Franck-Strasse, D-85747 Garching, Germany

Edited by Harden M. McConnell, Stanford University, Stanford, CA, and approved December 27, 2001 (received for review August 16, 2001) Magnetic tweezers are applied to study the enforced motion of filaments embedded in networks may be visualized and analyzed single actin filaments in entangled actin networks to gain insight by fluorescence microscopy (16) or by labeling with colloidal into friction-mediated entanglement in semiflexible macromolec- probes (17). In previous work, the latter possibility was used to ular networks. Magnetic beads are coupled to one chain end of test relate macroscopic viscoelastic impedance spectra to thermally filaments, which are pulled by 5 to 20 pN force pulses through driven single-filament motion. entangled solutions of nonlabeled actin, the test filaments thus The frequency-dependent viscoelastic impedance G*(␻) ϭ acting as linear force probes of the network. The transient filament GЈ(␻) ϩ iGЉ(␻) exhibits three distinct frequency regimes (18): at ␯ ϭ ␻͞ ␲ Ͼ ͞␶ ␶ Ͼ Ϫ2 ␶ motion is analyzed by microfluorescence, and the deflection- high frequencies, 2 1 e [ e 10 sec; note that e versus-time curves of the beads are evaluated in terms of a is the relaxation time of the bending mode of a chain segment of ⌳ mechanical equivalent circuit to determine viscoelastic parameters, length equal to the entanglement length e (6)], the shear elastic which are then interpreted in terms of viscoelastic moduli of the modulus GЈ(␻)Љ and the loss modulus GЉ(␻) increase with network. We demonstrate that the frictional coefficient character- frequency according to a power law GЈ(␻), GЉ(␻)ϰ␻0.75, which izing the hydrodynamic coupling of the filaments to the surround- has been shown to be determined by the entropic tension of the ing network is much higher than predicted by the tube model, single filaments (6, 8, 9, 11). The power law was verified suggesting that friction-mediated interfilament coupling plays an microscopically by analysis of the local motion of single filaments important role in the entanglement of non-cross-linked actin net- by using the colloidal probe technique (17). At medium fre- ͞␶ Ͻ ␯ Ͻ ͞␶ works. Furthermore, the local tube width along the filament quencies (1 d 1 e), a rubber-like plateau arises, and this contour (measured in terms of the root-mean-square displacement regime is determined by the affine shear deformation of the ␯ Ͻ ͞␶ characterizing the lateral of the test filament) network (11, 12). For 1 d, the entangled network becomes reveals strong fluctuations that can lead to transient local pinching fluid-like and the viscoelastic moduli decay to zero, whereas at ␯Ӎ ͞␶ ␶ Љ ␻ of filaments. 1 d ( d, terminal relaxation time), the loss modulus G ( ) exhibits a maximum. etworks of filamentous actin (F-actin) are of great interest Viscoelastic impedance spectra of the entangled actin network Nfrom the point of view of both cell biology and were calculated on the basis of the classical tube model (11, 12, physics and have thus been the subject of intense experimental 19, 20). Excellent agreement between the theoretical predictions and the spectra measured by torsional rheometry was found for and theoretical studies. On the one hand, actin is a major ␯ Ͼ ͞␶ the high-frequency regime ( 1 e), whereas at the low-

structural component of the intracellular scaffold (the cytoskel- CHEMISTRY frequency end of the plateau and within the terminal regime, the eton) and plays a key role in various cellular processes, such as theory underestimates the measured viscoelastic moduli. Thus, cell locomotion (1, 2), the transport processes within cells (3), or the measured friction coefficient is higher by a factor of 10 than the control of cell adhesion on surfaces (4). To fulfill this the theoretical prediction (see refs. 11 and 21), strongly suggest- multiplicity of tasks, nature uses a large number of helper ing that the tube model underestimates the frictional coupling proteins. These include severing proteins by which the length of between the filament and the environment. The failure of the actin filaments can be manipulated, monomer-sequestering pro- simple tube model also became evident by recent studies with the teins that allow the control of the polymer concentration, and colloidal probe technique showing that the tube diameter varies finally a manifold of cross-linker proteins enabling the genera- drastically (17). tion of randomly organized gels or arrangements of bundles To gain more detailed insight into the dynamic coupling acting as cell-stabilizing fibers or cables for intracellular trans- between single filaments and the surrounding chains, we applied port (5). the recently developed magnetic tweezer technique to study the On the other hand, F-actin is a prototype of a semiflexible enforced reptation of single test filaments. Magnetic beads (with polymer. Highly versatile models of entangled and cross-linked diameters smaller than mesh size) were attached to the end of networks of semiflexible macromolecules can be designed by phalloidin-stabilized and fluorescent-labeled actin filaments, controlling the structure through the manifold of manipulating which were embedded in entangled actin networks. These test proteins to study the distinct physical properties of this particular filaments were pulled through the network in a step-wise manner class of polymer networks. Such semiflexible polymer networks by application of sequences of force pulses on the magnetic exhibit outstanding viscoelastic features, which are determined by a subtle interplay of entropic and enthalpic contributions to the elastic free energy of the individual filament (6–12). This paper was submitted directly (Track II) to the PNAS office. One distinct advantage of actin as model polymer is the large Abbreviations: F-actin, filamentous actin; G-actin, monomeric actin. ␮ contour length (typically in the 10- m range) (13) and persis- † Ӎ ␮ To whom reprint requests should be addressed. E-mail: [email protected]. tence length (Lp 17 m) (10, 14, 15) enabling the design of The publication costs of this article were defrayed in part by page charge payment. This networks with mesh sizes in the optical wavelength regime. article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Therefore, the conformational dynamics and motion of single §1734 solely to indicate this fact.

www.pnas.org͞cgi͞doi͞10.1073͞pnas.052432499 PNAS ͉ May 14, 2002 ͉ vol. 99 ͉ no. 10 ͉ 6533–6538 Downloaded by guest on September 25, 2021 tweezer. The motion of the bead and the attached filament was analyzed by dynamic image processing. The filament motion induced by a force pulse consists of three regimes: first, a fast deflection with constant velocity associated with a deformation of the network; second, a slowing-down regime followed by flow of the filament with respect to the network; and third, partial relaxation associated with a backflow of the filament after switching off the force. The enforced filament motion is analyzed in terms of a mechanical equivalent circuit. The viscoelastic moduli obtained compare reasonably well with results of frequency-dependent measurements by torsional macrorheometry or magnetic bead microrheology. Distinct differences suggest, however, that the microviscoelastic moduli depend on the shape of the force probe and its interaction with the environment. A remarkable finding is that the frictional coupling between single filaments and the surrounding network is much stronger than predicted by the simple tube model of entangled macro- molecular networks. The heterogeneity of the entangled network was studied by pulling test filaments through the meshwork by sequences of force pulses. Very pronounced fluctuations of the effective tube diameter are observed that may even lead to the transient trapping of filaments in local narrows of the reptation tube, confirming previous findings that the tube diameter exhibits local narrows (17, 22). Materials and Methods The Magnetic Tweezers Setup. The experiments were performed with a previously described magnetic force microscope (23). This so-called magnetic tweezers setup allows application of se- Fig. 1. Typical response of test filaments to an applied external force. (a) quences of constant force pulses onto magnetic beads. It consists Image of a test filament embedded in an unlabeled actin network recorded by of a central measuring unit composed of a sample holder and one fluorescence microscopy. The magnetic bead was coupled to the fluorescent- magnetic coil (1,200 turns of 0.7-mm copper wire with an iron labeled filament at the right end. The superimposed arrow indicates the direction of the magnetic force F(t). (b) Two typical trajectories of the colloidal core exhibiting a sharp edge). This device is mounted on an beads attached to different test filaments are shown at high magnification AXIOVERT 10 microscope (Zeiss, Oberkochen, Germany). (Insets). Moreover, the filament contour before the application of the force The coil current is produced by a homemade voltage-controlled pulse is presented to show the overall length and curvature of the test current supply that transforms the voltage signal of a function filament. Note that the beads move not only along the x-axis (direction of the generator FG 9000 (ELV, Leer, Germany) into a current signal force) but also along the y-axis. with amplitudes of up to 4 A. The distance between the magnetic beads within the sample and the sharp edge-like tip of the iron core can be varied between 20 and 120 ␮m. The force-vs.- 2 mM DTT, and 5 mM ATP, pH 7.4). Biotin was coupled to distance relationship was calibrated following an established polymerized F-actin according to Okabe and Hirokawa (28). procedure (23) by measuring the velocity of the used magnetic Single fluorescent-labeled and streptavidin-coated paramag- beads in a water–glycerol solution of known for cur- netic beads (ProActive Streptavidin Superparamagnetic Classic, rents between 0.25 and 1.0 A. Both test filaments and magnetic Bangs Laboratories, Carmel, IN) with an average bead diameter beads were observed by classical fluorescence videomicroscopy r Ӎ 0.83 ␮m were coupled to the ends of single actin filaments by using a frame rate of 25 frames͞sec. by use of the well-known biotin–streptavidin system, as described (17). Finally, the bead-labeled actin filaments, which were Sample Preparation. Monomeric actin (so-called G-actin) was additionally fluorescent labeled and stabilized by rhodamine– prepared from rabbit skeletal muscle following the method of phalloidin, were embedded in a prepolymerized F-actin solution Pardee and Spudich (24). To remove residual cross-linking and following ref. 16. The concentration of the actin network was 0.2 capping proteins, it was purified by an additional step by using mg͞ml (5 ␮M) for all experiments corresponding to an average gel column chromatography (Sephacryl S-300) as described by network mesh size of ␨ ϭ 1.3 ␮m (see Fig. 1) to minimize direct MacLean-Fletcher and Pollard (25). According to previous interaction between the bead and its network surroundings. studies, residual cross-linkers are removed by this technique (26). Evidence for the absence of cross-linkers was further Data Acquisition. The method of data acquisition was described in provided by the finding that, in the viscoelastic spectra deter- detail in refs. 17 and 23. In brief: during the experiment, all mined for entangled networks of filamentous actin purified in observations were recorded on videotape by using a SIT-camera this way, the fluid-like terminal regime is well established (24). system (C2400, Hamamatsu, Herrsching, Germany). The se- G-actin was kept in G-buffer (consisting of 2 mM Imidazol, 0.2 quences of images were digitized by using a frame-grabber card mM CaCl2, 0.2 mM DTT, 0.5 mM ATP, and 0.005 vol-% NaN3, (PXC200, Imagination, Portland OR) combined with a personal pH ϭ 7.4) at 4°C and was used within 14 days of purification. The computer image-processing system. The spatial magnetic bead concentration of G-actin was determined by absorption spec- deflections were analyzed by a particle tracking method devel- troscopy assuming an extinction coefficient of 0.63 mgϪ1⅐mlϪ1 oped previously (17), enabling a time resolution of 40 msec and for absorption at 290 nm (27). Solutions of F-actin were prepared a spatial resolution of Ϯ2 nm in the image plane and of Ϯ200 nm by adding 1͞10 of the sample volume of 10-fold concentrated F along the optical axis. In addition, filament contours within the buffer (20 mM Imidazol, 1 M KCl, 2 mM CaCl2, 20 mM MgCl2, image plane were previously determined by a tracing algorithm

6534 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.052432499 Dichtl and Sackmann Downloaded by guest on September 25, 2021 In the short-time regime (I), the frictional stress is high, and the filament therefore induces a shear deformation of the surrounding network. After relaxation of the internal stresses within the network, the deformation saturates and the test filament moves with a constant velocity v1 relative to the network [corresponding to regime (II); see Fig. 2]. In the following, the viscoelastic response curves are analyzed in terms of the tube model by assuming that (because of the absence of crosslinkers) the test filaments are freely sliding, and their motion is impeded only by the frictional coupling to the surrounding network. This appears also to be justified by the finding in a separate series of experiments that the global ͐L Ѩ2ជ ͞Ѩ 2 2 curvature of the worm-like test filaments 0 ( r(s) s ) ds does not affect remarkably the viscoelastic parameters derived by the model discussed below.

Fig. 2. Mean normalized viscoelastic response curve ͗x˜(t)͘ϭ͗⌬x(t)͘͞F0 of a The initial phase of the deflection of the test filaments magnetic bead attached to a filament of length L ϭ 12.20 ␮m in an entangled (regime I) is determined by three contributions: (i) the stretch- network of mesh size ␰ ϭ 1.3 ␮m showing the short- (I) and long-time (II) ing of the wiggling filaments exhibiting excess length; (ii) the regime of the response and the partial relaxation (III). The amplitude of the elastic deflection of the network because of its frictional ϭ force pulse was F0 6.3 pN, and the duration 2.5 sec. Inset shows the simplest coupling with the filament; and (iii) the enforced reptation mechanical model, which can account for the observed viscoelastic response. motion of the locally stretched filament. To consider the first contribution, we measured the local mean square displacement (MSD) of the filaments by analyzing the Brownian motion of described previously (16). For the analysis of the viscoelastic small nonmagnetic beads (red fluorescent latex beads) at- response curves, we selected slightly curved filaments that were tached to test filaments by biotin–streptavidin linkers in the oriented with their long axes preferentially parallel to the direction parallel (͗⌬x2(t)͘) and perpendicular (͗⌬y2(t)͘)tothe direction of the applied magnetic force. local tube orientation. These force-free measurements were possible with 4-msec time resolution following the procedure Results and Discussion described earlier (17). At short times, the MSD of the bead- Phenomenology of Response Curves. Fig. 1 presents two typical labeled segments of the filament in both directions obeyed the examples of enforced filament reptation. Force pulses of scaling laws ͗⌬y2(t)͘ Ӎ ͗⌬x2(t)͘ϰt␣ with ␣ ϭ 0.75 Ϯ 0.06, which constant amplitude F0 and duration were applied to the is in agreement with theoretical predictions (10, 11, 29). After ␶ Ӎ ͗⌬ 2 ͘ filaments along the x axis. The transiently induced bead motion a relaxation time e 30 msec, the MSD y (t) saturates was recorded in the direction parallel x(t) and perpendicular because of the constraints imposed by the wall of the tube, y(t) to the force direction. The obvious broadening of the bead whereas the MSD ͗⌬x2(t)͘ exhibits a crossover into a linear trajectory perpendicular to the applied force is a consequence regime determined by the filament self diffusion along the tube of its local Brownian motion. This motion is shown at higher axis (17). At the present state of the microrheometry tech- magnification in Fig. 1b Insets. A notable result is that the bead nique, we can analyze the force-induced bead displacement trajectory exhibits a small component in the direction perpen- only with a time resolution of 40 msec, making it impossible to dicular to the force direction. For quantitative analysis, we resolve the contribution of the stretching regime of the test ͗⌬ ͘ϭ͗⌬ ͘͞ consider the viscoelastic response curves x˜(t) x(t) F0 filaments to the frictional coefficient discussed below. The of the beads, where ͗⌬x(t)͘ϭ͗x(t) Ϫ x (0)͘ is the mean stretching of the filaments, however, is clearly revealed by displacement of the bead in the field direction during an visual inspection with microfluorescence, which shows that the CHEMISTRY applied force pulse. In this way, the contribution of Brownian local wiggling motion is strongly restricted. In summary, our motion of the bead and the filament in the direction of the experiments suggest that the test filaments (exhibiting en- force is averaged out. The response curve ͗⌬x˜(t)͘ shown in Fig. forced reptation) can be considered as freely sliding threads 2 was obtained by averaging the response signal of a sequence that are smoothed by the frictional force, and that the filament of 15 force pulses applied to the test filament, which exhibits tension does not contribute to the elastic modulus introduced below (11). three distinct time regimes: (I) a rapid deflection with finite With the above considerations, one can describe this viscoelas- slope; (II) a retardation regime in which the bead slows down tic response by a mechanical equivalent circuit consisting of a and starts to move with nearly constant velocity; and finally, ␩ ␩ dashpot (viscosity 0), a parallel array of a dashpot (viscosity 1), (III) partial relaxation of the deflection after the force was and a spring (force constant ␮) often called a Kelvin–Voigt body switched off. ␩ (see Fig. 2 Inset). The dashpot 0 accounts for the long-time The enforced deformation of the actin network surrounding behavior (movement with constant velocity v1), whereas the the filament can be explained in terms of a strong initial Kelvin–Voigt body models the short-time response (movement frictional coupling between the test filament and the surround- with initial velocity v0 and retardation). The analysis of the ing actin chains. The filament moving with an initial velocity v0 response curves in terms of this equivalent circuit allows char- ␴ generates a hydrodynamic stress per unit length along the acterization of the entangled network in terms of viscoelastic contour of its cylindrical reptation tube of moduli per unit length of the test filament. The motional ␥ equation of the equivalent circuit for an applied force pulse F0 ˜v0 ␴ ϰ , [1] can be easily solved, yielding: ␰L 1 1 1 t ͗⌬ ͑ ͒͘ ϭ ͩ ͑ Ϫ Ϫt/␶͒ͪ ϩ ͩ ͪ where ␥˜ is an effective friction coefficient per unit length of the x˜ t ␮ 1 e ␩ , [2] g␮ g␩ 0 filament, ␰ is the average distance between the filament and the ␶ ϭ ␩ ͞␮ surrounding network, and L is the contour length of the test where the retardation time is 1 . The first term in Eq. 2 filament. accounts for the rapid deflection and relaxation of the internal

Dichtl and Sackmann PNAS ͉ May 14, 2002 ͉ vol. 99 ͉ no. 10 ͉ 6535 Downloaded by guest on September 25, 2021 stresses, whereas the second term describes the subsequent flow of the filament with respect to the network. The geometric factors g␮ and g␩ are introduced to relate the viscoelastic ␮ ␩ ␩ parameters , 1, and 0 to the frequency-dependent shear elastic modulus GЈ(␻), the loss modulus GЉ(␻) measured by ␩ oscillatory rheometry, and the zero-shear viscosity g of the entangled network derived from macrorheologic creep experiments. ϭ ␲ ͞␲ ␭ ͞ The value of g␩ 2 L n( h a) is obtained from the law of the frictional force ffrict on a cylinder of length L and an effective ␭ diameter a moving with velocity v in a tube of diameter h filled ␩ with a fluid of viscosity s (30): 2␲␩ L ϭ s ffrict ͑␭ ͞ ͒ v. [3] 1n h a ␭ According to the tube model of entangled polymer networks, h is the hydrodynamic screening length, which is about equal to the mesh size of the network ␰ (10) (a can be considered as an effective filament diameter to account for a residual dynamic roughness of the filament because of short-wavelength wiggling motion, because a is assumed to be small compared with the ␭ mesh size, and thus to h, the logarithmic correction would be small). Because we are not aware of a theory on the elastic defor- mation of a solid by an applied line force, we assume g␮ Ӎ g␩.As we consider pure shear deformation and because we deal with the situation of low Reynolds numbers, this approximation appears to be justified. In Fig. 3, we summarize values of the viscoelastic parameters Fig. 3. (a) Summary of shear elastic modulus times contour length gЈϭg␮␮ defined in Eqs. 2 and 3, which were obtained by the above (F, left ordinate) and retardation time ␶ (⅜, right ordinate) for test filaments Ӎ ␮ procedure for a selection of filaments with different contour of various contour lengths L in entangled networks of concentration ca 5 M ␰ Ӎ ␮ (mesh size ␰Ӎ1.3 ␮m). (b) Summary of frictional coefficients times contour lengths L in an entangled network of mesh size 1.3 m. Fig. Љϭ ␩ Јϭ ␮ length g g␩ 1 characterizing short-time response ( 224 , left ordinate) and 3a shows the values of the local network elasticity g g␮ , and ␨ ϭ ␩ ᮀ ␶ long-time viscous flow g g␩ 0 ( , right ordinate), respectively, for test the retardation time of the viscoelastic response. Fig. 3b shows filaments of various contour lengths L. The drawn line corresponds to a Љϭ ␩ the short-time frictional coefficient g g␩ 1 and the frictional least-square fit accounting for the linear increase of the viscoelasitic param- ␨ ϭ ␩ coefficient g g␩ 0, characterizing the flow of the filament eters with chain length. (regime II in Fig. 2). It is seen that the values fluctuate by about an order of magnitude, but that all moduli increase linearly with chain length as expected according to Eq. 3. The scattering of the sured at ␻͞2␲ Ն 1 Hz. Ruddies et al. (18) found a value of ␩ЈЈ data is a consequence of the large fluctuation of the local tube ϭ GЉ(␻)͞␻ Ӎ 0.15 Pa⅐sec at ␻͞2␲ ϭ 1 Hz, in reasonable width, which was already suggested by previous studies (17) and agreement with the present result. A nearly 10-fold smaller value which will be demonstrated below in a more direct way. of ␩ЉϷ0.03 Pa⅐sec has been found, however, between 1 and 40 To test first the reliability of our data analysis, it is useful to Hz by magnetic-bead microrheometry (32). compare the data of Fig. 3 with the viscoelastic moduli GЈ(␻) and GЉ(␻) of the frequency-dependent viscoelastic impedance Evaluation of Spatial Homogeneity of Network. To explore the G*(␻) measured for entangled actin solutions by macroscopic spatial homogeneity of the entangled network, we pulled test (10, 31) and microscopic rheometry (32). filaments through the network by a sequence of force pulses Consider first the microscopic elastic modulus ␮ ϭ gЈ͞g␮, and measured the local tube diameter along the path of the which is measured in units of Pa [such as GЈ(␻)]. With the filament. The example shown in Fig. 4 reveals a remarkable exception of the very long filament (L ϭ 25 ␮m), the values of heterogeneity of the network, resulting in three types of the shear modulus lie in the range 0.18 Յ ␮ Յ 0.58 Pa. These viscoelastic responses: for the first pulses, the response curve values compare well with the high-frequency storage moduli is similar to that of Fig. 2, consisting of an initial flow of the GЈ(␻͞2␲) measured recently by magnetic-bead microrheometry test filament followed by the induced viscoelastic response of (32) ranging from 0.15 Pa for 1 Hz to 0.3 Pa for 10 Hz (for the the surrounding network and a partial backflow of the filament comparison of data measured at different actin concentrations, (I); for the following five pulses, the test filament responds only ϰ 7͞5 ca we consider the scaling law G* ca (11). elastically but does not flow (II). After application of pulse 9, Consider now the short- and long-time friction: The long-time however, the test filament starts to flow again after a delay (III; ␩ ␩ ϭ ␨ ͞ viscosity g obtained from Fig. 3b according to g g g␩ varies note the parabolic shape of the response curves immediately between 3 and 30 Pa⅐sec. This large scattering of the values is after the onset of the force). For pulse 10, the behavior is the again attributed to the strong fluctuations of the local tube width. same as for of the first pulse. Obviously, the filament is pinched Nevertheless, the data are in reasonable agreement with mea- in a local trap during pulses 3–8. To gain insight into the nature surements of the zero-shear viscosity by torsional macrorheo- of the trap, we analyzed the local freedom of motion of the ␩ Ӎ ⅐ metry yielding g 13.00 Pa sec (31). filaments in the direction perpendicular to the average fila- ␩ ϭ Ј͞ ϭ ␶␮ The short-time defined as 1 g g␩ vary ments long axes before an applied force pulse. For that Յ ␩ Յ ⅐ between 0.16 1 0.49 Pa sec. Because theses values corre- purpose, microfluorescence images of the test filaments (typ- spond to the average retardation time of ␶ Ӎ 0.98 sec (see Fig. ically 250 images were taken at a frame rate of 25 images͞sec) 3a), the result has to be compared with rheological data mea- were superimposed and averaged. The resulting image con-

6536 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.052432499 Dichtl and Sackmann Downloaded by guest on September 25, 2021 Fig. 4. Demonstration of spatial inhomogeneities of entangled networks by measurement of local viscoelastic response curves of test filaments. The filament was pulled through the network by application of a sequence of force pulses (of amplitude F0 ϭ 8.8 pN). Note pinching of the filament during pulse nos. 3 and 8 and the delayed release from the trap during pulse number 9.

␩ tains, as a consequence, a direct visualization of the apparent compared with the solvent viscosity s and that the hydrody- reptation tube conformation (orientation and width) before an namic field generated by a point force decays as 1͞␩r rather ͞␩ applied force pulse. The fluorescence intensity distributions than as 1 sr. One therefore introduces a hydrodynamic ␭ perpendicular to the local filament axis were recorded. The screening length h (see Eq. 3). However, because the friction ␭ distributions could be well represented by Gaussians (35). The depends only logarithmically on h , the tube model cannot square root͌␴2 of the variance is a convenient measure of the account for the high apparent solvent viscosity measured ␰ ␭ local tube width (12). In Fig. 4 Insets, we show, for the unless h is approximately equal to the filament diameter. filament in nonpinched and a pinched states, respectively, the The agreement between theory and experiment is much variation of the tube width along the contours. Although in the better for viscoelastic impedance spectra measured by mac- former case the width ͌␴2 is constant along the whole roscopic torsional rheometry at high actin concentrations (Ӎ1 filament, the contour exhibits a narrowing over half of the mg͞ml). The apparent viscosity characterizing the filament filament length in the pinched state. Obviously, the friction motion obtained by comparison of experimental and theoret- ␩ ϭ ⅐ CHEMISTRY between the test filament and the adjacent network is so strong ical impedance spectra has been estimated at s 0.013 Pa sec that the test filament does not slip under the applied force. (12), which is larger by a factor of about 10 than the water However, repeated application of small stresses leads to the viscosity. The larger discrepancy between theory and experi- escape of the filament. This could be because of the local ment found by the present experiments corroborates previous softening of the networks mediated by either thermal fluctu- results obtained by colloidal bead microrheometry, suggesting ations or repeated action of forces below the local yield stress that the viscoelastic moduli measured by microrheometric of the network. The latter type of behavior has been recently techniques can differ drastically from values obtained by demonstrated for the transport of particles through cells by macrorheometry (32, 34, 37). In these studies, the storage and weak active forces and has been named viscoplasticity (36). loss moduli are underestimated, which is attributed to a depletion zone formed around the colloidal beads in the actin Concluding Discussion. We demonstrated that the enforced rep- network, as mentioned by Morse (11). The results found in the tation studies of single actin filaments by magnetic tweezers present study for linear force probes suggest that the momen- provide a useful tool to gain insight into the frictional coupling tum transfer of the test filament to the network is determined within entangled networks on a microscopic scale. The present by strong transient interfilament interactions, which can also study confirms previous microrheological work that provided be interpreted in terms of strong dynamic fluctuations of the evidence that the interpretation of viscoelastic data in terms tube diameter. Numerous electron microscope studies in the of the effective medium theory underestimates the frictional authors’ laboratory also suggest that several entangled fila- coupling between single filaments and the environment at long ments of random orientation have some tendency to converge times, that is, in the plateau and terminal regime (12). Thus, transiently to form parallel liquid crystal-like arrangements, ␩ ϭ ⅐ the smallest value of the long-time viscosity g 3Pasec is which can extend over a few microns before the filaments three orders of magnitude larger than the viscosity of water, separate again. The large value of the apparent solvent vis- whereas according to the tube model it should be equal to the cosity shows that the frictional coupling between the filaments solvent viscosity. In this model, one accounts for the hydro- can contribute strongly to the entanglement of semiflexible dynamic coupling by assuming that the medium surrounding macromolecular networks. the bead is a fluid exhibiting a constant viscosity ␩ large The strong dynamic fluctuations of the tube diameter sug-

Dichtl and Sackmann PNAS ͉ May 14, 2002 ͉ vol. 99 ͉ no. 10 ͉ 6537 Downloaded by guest on September 25, 2021 gested by the measurement of the frictional force on filaments finding that the pinched filaments become suddenly mobile again are corroborated by the pronounced spatial fluctuations of the (see Fig. 4) favors the idea that one deals with equilibrium tube width and the formation of local narrows (see Fig. 4). structures. Judging from our experiments, the average distance of such ␮ narrows is several 10 m. Such centers could thus also contribute Helpful discussions with A. Boulbitch, E. Frey, and K. Kroy are to the degree of entanglement of the network on a macroscopic gratefully acknowledged. This work was supported by the Deutsche scale. At present, we do not know whether the pinning centers Forschungsgemeinschaft (SFB 453) and the Fonds der Chemischen are in thermodynamic equilibrium with the bulk network. Our Industrie.

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