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

Abstract

Magnetic tweezers are applied to study the enforced motion of single actin filaments in entangled actin networks to gain insight into friction-mediated entanglement in semiflexible macromolecular networks. Magnetic beads are coupled to one chain end of test filaments, which are pulled by 5 to 20 pN force pulses through entangled solutions of nonlabeled actin, the test filaments thus acting as linear force probes of the network. The transient filament motion is analyzed by microfluorescence, and the deflection-versus-time curves of the beads are evaluated in terms of a mechanical equivalent circuit to determine viscoelastic parameters, which are then interpreted in terms of viscoelastic moduli of the network. We demonstrate that the frictional coefficient characterizing the hydrodynamic coupling of the filaments to the surrounding network is much higher than predicted by the tube model, suggesting that friction-mediated interfilament coupling plays an important role in the entanglement of non-cross-linked actin networks. Furthermore, the local tube width along the filament contour (measured in terms of the root-mean-square displacement characterizing the lateral Brownian motion of the test filament) reveals strong fluctuations that can lead to transient local pinching of filaments.

Networks of filamentous actin (F-actin) are of great interest from the point of view of both cell biology and polymer physics and have thus been the subject of intense experimental and theoretical studies. On the one hand, actin is a major structural component of the intracellular scaffold (the cytoskeleton) and plays a key role in various cellular processes, such as cell locomotion (1, 2), the transport processes within cells (3), or the control of cell adhesion on surfaces (4). To fulfill this multiplicity of tasks, nature uses a large number of helper proteins. These include severing proteins by which the length of actin filaments can be manipulated, monomer-sequestering proteins that allow the control of the polymer concentration, and finally a manifold of cross-linker proteins enabling the generation of randomly organized gels or arrangements of bundles acting as cell-stabilizing fibers or cables for intracellular transport (5).

On the other hand, F-actin is a prototype of a semiflexible polymer. Highly versatile models of entangled and cross-linked networks of semiflexible macromolecules can be designed by controlling the structure through the manifold of manipulating proteins to study the distinct physical properties of this particular class of polymer networks. Such semiflexible polymer networks 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).

One distinct advantage of actin as model polymer is the large contour length (typically in the 10-μm range) (13) and persistence length (L_p≃17 μm) (10, 14, 15) enabling the design of networks with mesh sizes in the optical wavelength regime. Therefore, the conformational dynamics and motion of single filaments embedded in networks may be visualized and analyzed by fluorescence microscopy (16) or by labeling with colloidal probes (17). In previous work, the latter possibility was used to relate macroscopic viscoelastic impedance spectra to thermally driven single-filament motion.

The frequency-dependent viscoelastic impedance G*(ω) = G′(ω) + iG"(ω) exhibits three distinct frequency regimes (18): at high frequencies, ν = ω/2π > 1/τ_e [τ_e > 10⁻² sec; note that τ_e is the relaxation time of the bending mode of a chain segment of length equal to the entanglement length Λ_e (6)], the shear elastic modulus G′(ω)" and the loss modulus G"(ω) increase with frequency according to a power law G′(ω), G"(ω)∝ω^0.75, which has been shown to be determined by the entropic tension of the single filaments (6, 8, 9, 11). The power law was verified microscopically by analysis of the local motion of single filaments by using the colloidal probe technique (17). At medium frequencies (1/τ_d < ν < 1/τ_e), a rubber-like plateau arises, and this regime is determined by the affine shear deformation of the network (11, 12). For ν < 1/τ_d, the entangled network becomes fluid-like and the viscoelastic moduli decay to zero, whereas at ν≃1/τ_d (τ_d, terminal relaxation time), the loss modulus G"(ω) exhibits a maximum.

Viscoelastic impedance spectra of the entangled actin network were calculated on the basis of the classical tube model (11, 12, 19, 20). Excellent agreement between the theoretical predictions and the spectra measured by torsional rheometry was found for the high-frequency regime (ν > 1/τ_e), whereas at the low-frequency end of the plateau and within the terminal regime, the theory underestimates the measured viscoelastic moduli. Thus, the measured friction coefficient is higher by a factor of 10 than the theoretical prediction (see refs. 11 and 21), strongly suggesting that the tube model underestimates the frictional coupling between the filament and the environment. The failure of the simple tube model also became evident by recent studies with the colloidal probe technique showing that the tube diameter varies drastically (17).

To gain more detailed insight into the dynamic coupling between single filaments and the surrounding chains, we applied the recently developed magnetic tweezer technique to study the enforced reptation of single test filaments. Magnetic beads (with diameters smaller than mesh size) were attached to the end of phalloidin-stabilized and fluorescent-labeled actin filaments, which were embedded in entangled actin networks. These test filaments were pulled through the network in a step-wise manner by application of sequences of force pulses on the magnetic 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 macromolecular 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 sequences of constant force pulses onto magnetic beads. It consists of a central measuring unit composed of a sample holder and one magnetic coil (1,200 turns of 0.7-mm copper wire with an iron core exhibiting a sharp edge). This device is mounted on an AXIOVERT 10 microscope (Zeiss, Oberkochen, Germany). The coil current is produced by a homemade voltage-controlled current supply that transforms the voltage signal of a function generator FG 9000 (ELV, Leer, Germany) into a current signal 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 μm. The force-vs.-distance relationship was calibrated following an established procedure (23) by measuring the velocity of the used magnetic beads in a water–glycerol solution of known viscosity for currents between 0.25 and 1.0 A. Both test filaments and magnetic beads were observed by classical fluorescence videomicroscopy by using a frame rate of 25 frames/sec.

Sample Preparation.

Monomeric actin (so-called G-actin) was prepared from rabbit skeletal muscle following the method of Pardee and Spudich (24). To remove residual cross-linking and capping proteins, it was purified by an additional step by using gel column chromatography (Sephacryl S-300) as described by MacLean-Fletcher and Pollard (25). According to previous studies, residual cross-linkers are removed by this technique (26). Evidence for the absence of cross-linkers was further provided by the finding that, in the viscoelastic spectra determined for entangled networks of filamentous actin purified in this way, the fluid-like terminal regime is well established (24). G-actin was kept in G-buffer (consisting of 2 mM Imidazol, 0.2 mM CaCl₂, 0.2 mM DTT, 0.5 mM ATP, and 0.005 vol-% NaN₃, pH = 7.4) at 4°C and was used within 14 days of purification. The concentration of G-actin was determined by absorption spectroscopy assuming an extinction coefficient of 0.63 mg⁻¹⋅ml⁻¹ for absorption at 290 nm (27). Solutions of F-actin were prepared by adding 1/10 of the sample volume of 10-fold concentrated F buffer (20 mM Imidazol, 1 M KCl, 2 mM CaCl₂, 20 mM MgCl₂, 2 mM DTT, and 5 mM ATP, pH 7.4). Biotin was coupled to polymerized F-actin according to Okabe and Hirokawa (28). Single fluorescent-labeled and streptavidin-coated paramagnetic beads (ProActive Streptavidin Superparamagnetic Classic, Bangs Laboratories, Carmel, IN) with an average bead diameter r ≃ 0.83 μm were coupled to the ends of single actin filaments by use of the well-known biotin–streptavidin system, as described (17). Finally, the bead-labeled actin filaments, which were additionally fluorescent labeled and stabilized by rhodamine–phalloidin, were embedded in a prepolymerized F-actin solution following ref. 16. The concentration of the actin network was 0.2 mg/ml (5 μM) for all experiments corresponding to an average network mesh size of ζ = 1.3 μm (see Fig. 1) to minimize direct interaction between the bead and its network surroundings.

Figure 1.

Typical response of test filaments to an applied external force. (a) Image of a test filament embedded in an unlabeled actin network recorded by fluorescence microscopy. The magnetic bead was coupled to the fluorescent- 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 beads attached to different test filaments are shown at high magnification (Insets). Moreover, the filament contour before the application of the force pulse is presented to show the overall length and curvature of the test filament. Note that the beads move not only along the x-axis (direction of the force) but also along the y-axis.

Data Acquisition.

The method of data acquisition was described in detail in refs. 17 and 23. In brief: during the experiment, all observations were recorded on videotape by using a SIT-camera system (C2400, Hamamatsu, Herrsching, Germany). The sequences of images were digitized by using a frame-grabber card (PXC200, Imagination, Portland OR) combined with a personal computer image-processing system. The spatial magnetic bead deflections were analyzed by a particle tracking method developed previously (17), enabling a time resolution of 40 msec and a spatial resolution of ±2 nm in the image plane and of ±200 nm along the optical axis. In addition, filament contours within the image plane were previously determined by a tracing algorithm described previously (16). For the analysis of the viscoelastic response curves, we selected slightly curved filaments that were oriented with their long axes preferentially parallel to the direction of the applied magnetic force.

Results and Discussion

Phenomenology of Response Curves.

Fig. 1 presents two typical examples of enforced filament reptation. Force pulses of constant amplitude F₀ and duration were applied to the filaments along the x axis. The transiently induced bead motion was recorded in the direction parallel x(t) and perpendicular y(t) to the force direction. The obvious broadening of the bead trajectory perpendicular to the applied force is a consequence of its local Brownian motion. This motion is shown at higher magnification in Fig. 1b Insets. A notable result is that the bead trajectory exhibits a small component in the direction perpendicular to the force direction. For quantitative analysis, we consider the viscoelastic response curves 〈Δx̃(t)〉 = 〈Δx(t)〉/F₀ of the beads, where 〈Δx(t)〉 = 〈x(t) − x (0)〉 is the mean displacement of the bead in the field direction during an applied force pulse. In this way, the contribution of Brownian motion of the bead and the filament in the direction of the force is averaged out. The response curve 〈Δx̃(t)〉 shown in Fig. 2 was obtained by averaging the response signal of a sequence of 15 force pulses applied to the test filament, which exhibits three distinct time regimes: (I) a rapid deflection with finite slope; (II) a retardation regime in which the bead slows down and starts to move with nearly constant velocity; and finally, (III) partial relaxation of the deflection after the force was switched off.

Figure 2.

Mean normalized viscoelastic response curve 〈x̃(t)〉 = 〈Δx(t)〉/F₀ of a magnetic bead attached to a filament of length L = 12.20 μm in an entangled network of mesh size ξ = 1.3 μm showing the short- (I) and long-time (II) regime of the response and the partial relaxation (III). The amplitude of the force pulse was F₀ = 6.3 pN, and the duration 2.5 sec. Inset shows the simplest mechanical model, which can account for the observed viscoelastic response.

The enforced deformation of the actin network surrounding the filament can be explained in terms of a strong initial frictional coupling between the test filament and the surrounding actin chains. The filament moving with an initial velocity v₀ generates a hydrodynamic stress σ per unit length along the contour of its cylindrical reptation tube of

1

where γ̃ is an effective friction coefficient per unit length of the filament, ξ is the average distance between the filament and the surrounding network, and L is the contour length of the test filament.

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 curvature of the worm-like test filaments ∫(∂²r⃗(s)/∂s²)²ds does not affect remarkably the viscoelastic parameters derived by the model discussed below.

The initial phase of the deflection of the test filaments (regime I) is determined by three contributions: (i) the stretching of the wiggling filaments exhibiting excess length; (ii) the elastic deflection of the network because of its frictional coupling with the filament; and (iii) the enforced reptation 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 small nonmagnetic beads (red fluorescent latex beads) attached to test filaments by biotin–streptavidin linkers in the direction parallel (〈Δx²(t)〉) and perpendicular (〈Δy²(t)〉) to the local tube orientation. These force-free measurements were possible with 4-msec time resolution following the procedure described earlier (17). At short times, the MSD of the bead-labeled segments of the filament in both directions obeyed the scaling laws 〈Δy²(t)〉 ≃ 〈Δx²(t)〉∝t^α with α = 0.75 ± 0.06, which is in agreement with theoretical predictions (10, 11, 29). After a relaxation time τ_e≃30 msec, the MSD 〈Δy²(t)〉 saturates because of the constraints imposed by the wall of the tube, whereas the MSD 〈Δx²(t)〉 exhibits a crossover into a linear regime determined by the filament self diffusion along the tube axis (17). At the present state of the microrheometry technique, we can analyze the force-induced bead displacement only with a time resolution of 40 msec, making it impossible to resolve the contribution of the stretching regime of the test filaments to the frictional coefficient discussed below. The stretching of the filaments, however, is clearly revealed by visual inspection with microfluorescence, which shows that the local wiggling motion is strongly restricted. In summary, our experiments suggest that the test filaments (exhibiting enforced reptation) can be considered as freely sliding threads that are smoothed by the frictional force, and that the filament tension does not contribute to the elastic modulus introduced below (11).

With the above considerations, one can describe this viscoelastic response by a mechanical equivalent circuit consisting of a dashpot (viscosity η₀), a parallel array of a dashpot (viscosity η₁), and a spring (force constant μ) often called a Kelvin–Voigt body (see Fig. 2 Inset). The dashpot η₀ accounts for the long-time behavior (movement with constant velocity v₁), whereas the Kelvin–Voigt body models the short-time response (movement with initial velocity v₀ and retardation). The analysis of the response curves in terms of this equivalent circuit allows characterization of the entangled network in terms of viscoelastic moduli per unit length of the test filament. The motional equation of the equivalent circuit for an applied force pulse F₀ can be easily solved, yielding:

2

where the retardation time is τ = η₁/μ. The first term in Eq. 2 accounts for the rapid deflection and relaxation of the internal stresses, whereas the second term describes the subsequent flow of the filament with respect to the network. The geometric factors g_μ and g_η are introduced to relate the viscoelastic parameters μ, η₁, and η₀ to the frequency-dependent shear elastic modulus G′(ω), the loss modulus G"(ω) measured by oscillatory rheometry, and the zero-shear viscosity η_g of the entangled network derived from macrorheologic creep experiments.

The value of g_η = 2πL/πn(λ_h/a) is obtained from the law of the frictional force f_frict on a cylinder of length L and an effective diameter a moving with velocity v in a tube of diameter λ_h filled with a fluid of viscosity η_s (30):

3

According to the tube model of entangled polymer networks, λ_h is the hydrodynamic screening length, which is about equal to the mesh size of the network ξ (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 λ_h, the logarithmic correction would be small).

Because we are not aware of a theory on the elastic deformation of a solid by an applied line force, we assume g_μ ≃ g_η. 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 defined in Eqs. 2 and 3, which were obtained by the above procedure for a selection of filaments with different contour lengths L in an entangled network of mesh size ξ ≃ 1.3 μm. Fig. 3a shows the values of the local network elasticity g′ = g_μμ, and the retardation time τ of the viscoelastic response. Fig. 3b shows the short-time frictional coefficient g" = g_ηη₁ and the frictional coefficient ζ_g = g_ηη₀, characterizing the flow of the filament (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 data is a consequence of the large fluctuation of the local tube width, which was already suggested by previous studies (17) and which will be demonstrated below in a more direct way.

Figure 3.

(a) Summary of shear elastic modulus times contour length g′ = g_μμ (●, left ordinate) and retardation time τ (○, right ordinate) for test filaments of various contour lengths L in entangled networks of concentration c_a ≃5 μM (mesh size ξ≃1.3 μm). (b) Summary of frictional coefficients times contour length g" = g_ηη₁ characterizing short-time response ( 224 , left ordinate) and long-time viscous flow ζ_g = g_ηη₀ (□, right ordinate), respectively, for test filaments of various contour lengths L. The drawn line corresponds to a least-square fit accounting for the linear increase of the viscoelasitic parameters with chain length.

To test first the reliability of our data analysis, it is useful to compare the data of Fig. 3 with the viscoelastic moduli G′(ω) and G"(ω) of the frequency-dependent viscoelastic impedance G*(ω) measured for entangled actin solutions by macroscopic (10, 31) and microscopic rheometry (32).

Consider first the microscopic elastic modulus μ = g′/g_μ, which is measured in units of Pa [such as G′(ω)]. With the exception of the very long filament (L = 25 μm), the values of the shear modulus lie in the range 0.18 ≤ μ ≤ 0.58 Pa. These values compare well with the high-frequency storage moduli G′(ω/2π) measured recently by magnetic-bead microrheometry (32) ranging from 0.15 Pa for 1 Hz to 0.3 Pa for 10 Hz (for the comparison of data measured at different actin concentrations, c_a we consider the scaling law G*∝c (11).

Consider now the short- and long-time friction: The long-time viscosity η_g obtained from Fig. 3b according to η_g = ζ_g/g_η varies between 3 and 30 Pa⋅sec. This large scattering of the values is again attributed to the strong fluctuations of the local tube width. Nevertheless, the data are in reasonable agreement with measurements of the zero-shear viscosity by torsional macrorheometry yielding η_g ≃ 13.00 Pa⋅sec (31).

The short-time viscosities defined as η₁ = g′/g_η = τμ vary between 0.16 ≤ η₁ ≤ 0.49 Pa⋅sec. Because theses values correspond to the average retardation time of τ ≃ 0.98 sec (see Fig. 3a), the result has to be compared with rheological data measured at ω/2π ≥ 1 Hz. Ruddies et al. (18) found a value of η′′ = G"(ω)/ω ≃ 0.15 Pa⋅sec at ω/2π = 1 Hz, in reasonable agreement with the present result. A nearly 10-fold smaller value of η" ≈0.03 Pa⋅sec has been found, however, between 1 and 40 Hz by magnetic-bead microrheometry (32).

Evaluation of Spatial Homogeneity of Network.

To explore the spatial homogeneity of the entangled network, we pulled test filaments through the network by a sequence of force pulses and measured the local tube diameter along the path of the filament. The example shown in Fig. 4 reveals a remarkable heterogeneity of the network, resulting in three types of viscoelastic responses: for the first pulses, the response curve is similar to that of Fig. 2, consisting of an initial flow of the test filament followed by the induced viscoelastic response of the surrounding network and a partial backflow of the filament (I); for the following five pulses, the test filament responds only elastically but does not flow (II). After application of pulse 9, however, the test filament starts to flow again after a delay (III; note the parabolic shape of the response curves immediately after the onset of the force). For pulse 10, the behavior is the same as for of the first pulse. Obviously, the filament is pinched in a local trap during pulses 3–8. To gain insight into the nature of the trap, we analyzed the local freedom of motion of the filaments in the direction perpendicular to the average filaments long axes before an applied force pulse. For that purpose, microfluorescence images of the test filaments (typically 250 images were taken at a frame rate of 25 images/sec) were superimposed and averaged. The resulting image contains, as a consequence, a direct visualization of the apparent reptation tube conformation (orientation and width) before an applied force pulse. The fluorescence intensity distributions perpendicular to the local filament axis were recorded. The distributions could be well represented by Gaussians (35). The square root of the variance is a convenient measure of the local tube width ξ (12). In Fig. 4 Insets, we show, for the filament in nonpinched and a pinched states, respectively, the variation of the tube width along the contours. Although in the former case the width is constant along the whole filament, the contour exhibits a narrowing over half of the filament length in the pinched state. Obviously, the friction between the test filament and the adjacent network is so strong that the test filament does not slip under the applied force. However, repeated application of small stresses leads to the escape of the filament. This could be because of the local softening of the networks mediated by either thermal fluctuations or repeated action of forces below the local yield stress of the network. The latter type of behavior has been recently demonstrated for the transport of particles through cells by weak active forces and has been named viscoplasticity (36).

Figure 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 = 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.

Concluding Discussion.

We demonstrated that the enforced reptation studies of single actin filaments by magnetic tweezers provide a useful tool to gain insight into the frictional coupling within entangled networks on a microscopic scale. The present study confirms previous microrheological work that provided evidence that the interpretation of viscoelastic data in terms of the effective medium theory underestimates the frictional coupling between single filaments and the environment at long times, that is, in the plateau and terminal regime (12). Thus, the smallest value of the long-time viscosity η_g = 3 Pa⋅sec is three orders of magnitude larger than the viscosity of water, whereas according to the tube model it should be equal to the solvent viscosity. In this model, one accounts for the hydrodynamic coupling by assuming that the medium surrounding the bead is a fluid exhibiting a constant viscosity η large compared with the solvent viscosity η_s and that the hydrodynamic field generated by a point force decays as 1/ηr rather than as 1/η_sr. One therefore introduces a hydrodynamic screening length λ_h (see Eq. 3). However, because the friction depends only logarithmically on λ_h , the tube model cannot account for the high apparent solvent viscosity measured unless λ_h is approximately equal to the filament diameter.

The agreement between theory and experiment is much better for viscoelastic impedance spectra measured by macroscopic torsional rheometry at high actin concentrations (≃1 mg/ml). The apparent viscosity characterizing the filament motion obtained by comparison of experimental and theoretical impedance spectra has been estimated at η_s = 0.013 Pa⋅sec (12), which is larger by a factor of about 10 than the water viscosity. The larger discrepancy between theory and experiment found by the present experiments corroborates previous results obtained by colloidal bead microrheometry, suggesting that the viscoelastic moduli measured by microrheometric techniques can differ drastically from values obtained by macrorheometry (32, 34, 37). In these studies, the storage and loss moduli are underestimated, which is attributed to a depletion zone formed around the colloidal beads in the actin network, as mentioned by Morse (11). The results found in the present study for linear force probes suggest that the momentum transfer of the test filament to the network is determined by strong transient interfilament interactions, which can also be interpreted in terms of strong dynamic fluctuations of the tube diameter. Numerous electron microscope studies in the authors' laboratory also suggest that several entangled filaments of random orientation have some tendency to converge transiently to form parallel liquid crystal-like arrangements, which can extend over a few microns before the filaments separate again. The large value of the apparent solvent viscosity shows that the frictional coupling between the filaments can contribute strongly to the entanglement of semiflexible macromolecular networks.

The strong dynamic fluctuations of the tube diameter suggested by the measurement of the frictional force on filaments are corroborated by the pronounced spatial fluctuations of the tube width and the formation of local narrows (see Fig. 4). Judging from our experiments, the average distance of such narrows is several 10 μm. Such centers could thus also contribute to the degree of entanglement of the network on a macroscopic scale. At present, we do not know whether the pinning centers are in thermodynamic equilibrium with the bulk network. Our finding that the pinched filaments become suddenly mobile again (see Fig. 4) favors the idea that one deals with equilibrium structures.

Abbreviations

F-actin

filamentous actin

G-actin

monomeric actin