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Single-Filament Dynamics and Long-Range Ordering of Semiflexible Biopolymers under Flow and Confinement Laurent Vonna,*,† Laurent Limozin,‡ Alexander Roth,§ and Erich Sackmann§ Institut de Chimie des Surfaces et Interfaces (UPR 9069), 15 rue Jean Starcky - BP2478-68057 Mulhouse Cedex, France, Hoˆ pital Ste Marguerite, Laboratoire d’Immunologie (Inserm U600 CNRS FRE 2059), BP 29, 13274 Marseille Cedex 09, France, and Physik Depatment E22 (biophysics group), Technische Universita¨ t Mu¨ nchen, D-85748 Garching, Germany Received April 13, 2005. In Final Form: June 7, 2005 We report the collective and single-filament dynamics of long semiflexible actin filaments flowing in an evaporating droplet adhering on glass and accumulating along the physical barrier constituted by the droplet triple line. The observation of fluorescent reporter filaments embedded in the entangled network enables us to relate the final collective organization of the accumulated filaments to the individual filament dynamics. Three areas corresponding to distinct filament organizations are observed in the region of the initial triple line pinning, after complete evaporation of the droplet. A nematic liquid-crystal-like alignment of the filaments is observed at the edge of the droplet because of the dynamic filament alignment, whereas a less-ordered packing is generated because of the bending and folding of most of the filaments. The latter unconventional dynamics is analyzed in terms of the amplification of undulation modes typical of semiflexible polymers. The receding regime of the droplet triple line leads finally to a remaining film of actin filaments showing random organization.
Introduction The controlled organization of nanoparticles on mesoscopic scales is of major interest in many fields. For example, electronic or optical material properties can be optimized by self-assembly of metallic nanoparticles.1 The cohesiveness of latex films can be improved through ordering of nanoscopic latex particles.2 Specific adhesive or bioadhesive properties can be generated by distinct organizations of polymers or biopolymers at interfaces.3 The structural organization can be controlled by the interplay of specific or nonspecific interactions between particles or molecules or by the application of an external force. Such forces can be mediated by hydrodynamic shear or capillary forces,4,5 electric fields,6 magnetic fields,7 or centrifugal forces in the case of spin coating.8 A different strategy is to make use of steric confinement or a surface topography that has been shown to enable the distinct organization of macromolecules.9,10 Interestingly, an evaporating droplet presents both confinement and flow features. The evaporation combined with the pinning of * Corresponding author. E-mail:
[email protected]. † Institut de Chimie des Surfaces et Interfaces (UPR 9069). ‡ Ho ˆ pital Ste Marguerite. § Technische Universita ¨ t Mu¨nchen. (1) Murray, C. B.; Kagan, C. R.; Bawendi, M. G. Science 1995, 270, 1335. (2) Keddie, J. L. Mater. Sci. Eng. 1997, 21, 101. (3) Burmeister, J.; Vrany, J.; Reichert, W.; Truskey, G. J. Biomed. Mater. Res. 1996, 30, 13. (4) Dimitrov, A. S.; Dushkin, C. D.; Yoshimura, H.; Nagayama, K. Langmuir 1994, 10, 432. (5) Denkov, N.; Velev, O.; Kralchevski, P.; Ivanov, I.; Yoshimura, H.; Nagayama, K. Langmuir 1992, 8, 3183. (6) Bakajin, O. B.; Duke, T. A. J.; Chou, C. F.; Chan, S. S.; Austin, R. H.; Cox, E. C. Phys. Rev. Lett. 1998, 80, 2737. (7) Dimitrov, A. S.; Takahashi, T.; Furusawa, K.; Nagayama, K. J. Phys. Chem. 1996, 100, 3163. (8) Yokota, H.; Sunwoo, J.; Sarikaya, M.; van den Engh, G.; Aebersold, R. Anal. Chem. 1999, 71, 4418. (9) Fasolka, M. J.; Harris, D. J.; Mayes, A. M.; Yoon, M.; Mochrie, S. G. J. Phys. Rev. Lett. 1997, 79, 3018. (10) Rockford, L.; Liu, Y.; Mansky, P.; Russell, T. P.; Yoon, M.; Mochrie, S. G. J. Phys. Rev. Lett. 1999, 82, 2602.
the triple line induced an outward flow that accumulates the solute in the confined space materialized by the wedge.11 In this configuration, it is possible to control the growth of long-range organized domains of particles such as polystyrene microbeads12-14 or metallic nanoparticles.15,16 Until now, most of these experiments have been concerned with the organization of rigid spherical-like objects. Only a few recent works have described the behavior of semiflexible macromolecules in an evaporating droplet.17,18 We report in this article the behavior of semiflexible polymers solutions under the condition of flow and confinement mediated by gradually evaporating droplets. We choose for this work actin as a model polymer, which is a semiflexible biopolymer with a diameter of 7 nm and a persistence length around 18 µm.19-21 It is abundant in mammalian cells either as a globular monomer (called G-actin) or as a filament (called F-actin) and in total represents between 10 and 20% of the total mass of cytoplasmic proteins. Many intra- and extracellular signals control the polymerization of G-actin to form F-actin. In combination with other cytoplasmic proteins, the polymerized actin can generate a multitude of mesostructures that form the machinery for the generation (11) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Nature 1997, 389, 827. (12) Adachi, E.; Dimitrov, A. S.; Nagayama, K. Langmuir 1995, 11, 1057. (13) Cademartiri, L.; Sutti, A.; Calestani, G.; Dionigi, C.; Nozar, P.; Migliori, A. Langmuir 2003, 19, 7944. (14) Fischer, B. J. Langmuir 2002, 18, 60. (15) Mougin, K.; Haidara, H. Langmuir 2002, 18, 9566. (16) Maenosono, S.; Dushkin, C. D.; Saita, S.; Yamaguchi, Y. Langmuir 1999, 15, 957. (17) Pfohl, T.; Kim, J. H.; Yasa, M.; Miller, H. P.; Wong, G. C. L.; Bringezu, F.; Wen, Z.; Wilson, L.; Kim, M. W.; Li, Y.; Safinya, C. R. Langmuir 2001, 17, 5343. (18) Kimura, M.; Misner, M. J.; Xu, T.; Kim, S. H.; Russell, T. P. Langmuir 2003, 19, 9910. (19) Ka¨s, J.; Strey, H.; Sackmann, E. Nature 1994, 368, 226. (20) Ott, A.; Magnasco, M.; Simon, A.; Libchaber, A. Phys. Rev. A 1993, 48, R1642. (21) Hinner, B.; Tempel, M.; Sackmann, E.; Kroy, K.; Frey, E. Phys. Rev. Lett. 1998, 81, 2614.
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of mechanical forces in chemomechanical processes, such as locomotion22 or phagocytosis.23 These structures include bundles (aligned filaments) in filopodia or stress fibers or sheets (crossed filaments) in lamellipodia. In the present work, a droplet of an aqueous actin filaments solution is deposited onto a glass substrate. Solvent evaporation and the pinning of the triple line induces an outward hydrodynamic flow that pushes the actin filaments toward the edge of the droplet. By doping the actin solution with fluorescent filaments, we could observe the microscopic organization of the actin network as well as the single and collective filament dynamics during the flowing and packing process. At the edge of the droplet filaments, nematic-like structures with the direction oriented parallel to the triple line are formed. Toward the center of the droplet, a film of actin filaments with more random organization remains on the substrate only after the detachment and the receding of the triple line. Our results complement former studies of particles organization in the wedge of an evaporating droplet giving insight into the distribution of a solute in an evaporating system, which is an important phenomena in many industrial processes. Compared to the behavior of flexible polymers in a flow, the actin network in the evaporating droplet shows a number of distinct and remarkable features. These include the orientation of the whole actin network in the direction of the fluid flow, the collective alignment of macromolecules flowing against a physical barrier oriented perpendicular to the flow, and the folding and bending of the filaments in this wedge-shaped confined geometry. Our experiment shows that a proper coupling of confinement and flow can induce an organized arrangement of semiflexible biopolymers. The process suggests an alternative and biologically relevant strategy to techniques normally used to align collagen fibrils, for example, the stretch-induced orientation under a magnetic field,24 convection,25 or drainage.26,27 Such supramolecular organizations are of fundamental importance for the evolution of a biological system because they were shown to control cell adhesion or to induce contact guidance.26,28-31 Experimental Section Actin was prepared in the laboratory from rabbit skeletal muscles following a procedure described earlier in the literature32 and stored in monomeric form (G-actin) in G-buffer (2 mM tris/ HCl pH 7.5, 0.2 mM ATP, 0.2 mM CaCl2, 0.2 mM dithiothreitol, 0.005% NaN3) at a temperature of 4 °C. The filamentous actin (F-actin) was polymerized from the monomer G-actin solution by adding Mg2+ (2 mM), and polymerization occurred for 1 h at room temperature. The fluorescent actin filaments were labeled with rhodamine-phalloidin (Sigma) in a ratio of 1:1 phalloidin to actin monomer concentration. Both fluorescent and nonfluorescent actin filament solutions are gently mixed before each experiments in a filament ratio of 1:1000. The physical properties of the filaments and the actin solution are determined in parallel studies because they depend on the preparation procedure and the storage conditions.33 The polymerization conditions chosen in this work lead to filaments with (22) Mitchison, T. J.; Cramer, L. P. Cell 1996, 84, 371. (23) Chimini, G.; Chavrier, P. Nat. Cell Biol. 2000, 2, 191. (24) Torbet, J.; Ronziere, M. C. Biochem. J. 1984, 219, 1057. (25) Ghosh, S.; Comper, W. D. Connect. Tissue Res. 1988, 17, 33. (26) Dunn, G. A.; Ebendal, T. Exp. Cell Res. 1978, 111, 475. (27) Elsdale, T.; Bard, J. J. Cell Biol. 1972, 54, 626. (28) Guido, S.; Tranquillo, R. T. J. Cell Sci. 1993, 105, 317. (29) Boocock, C. A. Development 1989, 107, 881. (30) Zhua, B.; Zhangb, Q.; Lua, Q.; Xub, Y.; Yina, J.; Hub, J.; Wang, Z. Biomaterials 2004, 25, 4215. (31) Dickinson, R. B.; Guido, S.; Tranquillo, R. T. Ann. Biomed. Eng. 1994, 22, 342. (32) Pardee, J. D.; Spudich, J. A. Methods Enzymol. 1982, 85, 164.
Vonna et al. an average contour length of L ≈ 23 ( 17 µm,34,35 similar to what is described in the literature.36 The homogeneity of the actin solution was checked by electron microscopy,34 showing that at the concentration used in this work homogeneous entangled networks are formed and filaments rarely aggregated. This was also verified by fluorescence microscopy, which is presented below. The actin monomer concentrations used in our case (ranging from c ) 2.5 to 20 µM) correspond to a mesh size between ξ ) 1.05 and 0.37 µm according to the relation ξ ) 1.6c-1/2. Because the average length of the filaments (∼20 µm) is large compared to the entanglement length (which is on the order of the mesh size, ∼ξ), the network is strongly entangled. Finally, both viscoelastic parameterssthe loss and storage moduli G′(ω) and G′′(ω), respectivelyswere determined by magnetic tweezer microrheometry.37 The persistence length of the fluorescent filaments was not measured for our buffer conditions but can be considered to be around Lp ≈ 18 µm according to the literature.38 To measure the flow of the fluid separately from that of the actin filaments, we suspended in some experiments 40-nmdiameter fluorescent microbeads from Molecular Probes (cat. no. F8780). The beads were dispersed in a similar buffer to that used for actin, which was finally mixed with the actin solution in an appropriate amount. For each experiment, a droplet of F-actin is settled on a glass substrate cleaned with acetone and ethanol in an ultrasound bath without any further treatment. The edge of the droplet is immediately observed with an Axiovert 200 Zeiss microscope using a 63× antiflex objective (Zeiss, Oberkochen, Germany) that enables us to switch rapidly between the reflection interference contrast mode (RICM) and the fluorescence mode (excitation wavelength λ ) 546 nm for each mode). The evolution of the droplet edge and the dynamics of the filaments are recorded with a CCD camera connected to a numerical acquisition system, and the tracking of the fluorescent beads is performed with homemade software.
Results A droplet of F-actin solution is deposited onto a clean glass substrate. The subsequent dynamics of wetting and dewetting of the droplet is observed by reflection interference contrast microscopy (RICM), whereas the individual filaments are observed in fluorescence mode. 1. Drop Profile. First, the droplet spreads for some time, and the triple line is eventually pinned. The contact angle is approximately determined by RICM images as shown in Figure 1.39 The RICM image of the partially wetting droplet exhibits arrangements of interference fringes near the triple line. From the distances of the fringes, the profile of the droplet close to the triple line can be estimated. The height difference h between two minima (dark fringes) is given by h ) λ/2nw ) 203 nm (with λ ) 546 nm being the wavelength and nw being the refractive index of water). Figure 1 shows the distribution of the light intensity of a section of the wedge perpendicular to the triple line (black line on the images), whereas the reconstructed profile of the wedge of the droplet is shown in the inset. The microscopic equilibrium wetting contact angle in the particular case of Figure 1a is about 13°. The high affinity between the aqueous solution and the hydrophilic glass surface leads to a strong spreading of (33) Xu, J.; Schwarz, W. H.; Ka¨s, J. A.; Stossel, T. P.; Janmey, P. A.; Pollard, T. D. Biophys. J. 1998, 74, 2731. (34) Limozin, L.; Ba¨rmann, M.; Sackmann, E. Eur. Phys. J. E 2003, 10, 319. (35) Kaufmann, S.; Kas, J.; Goldmann, W. H.; Sackmann, E.; Isenberg, G. FEBS Lett. 1992, 314, 203. (36) Kas, J.; Strey, H.; Tang, J. X.; Finger, D.; Ezzell, R.; Sackmann, E.; Janmey, P. A. Biophys. J. 1996, 70, 609. (37) Schmidt, F. G.; Hinner, B.; Sackmann, E. Phys. Rev. E 2000, 61, 5646. (38) Isambert, H.; Venier, P.; Maggs, A. C.; Fattoum, A.; Kassab, R.; Pantaloni, D.; Carlier, M. F. J. Biol. Chem. 1995, 270, 11437. (39) Wiegand, G.; Neumaier, K. R.; Sackmann, E. Appl. Opt. 1998, 3729, 6892.
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Figure 1. Evolution of the droplet wedge observed with RICM during the initial stage of the drying process. For each image, the intensity distribution along section AB is plotted on the right side. The reconstruction of the wedge profile corresponding to this distribution is shown in the insets. It is possible to distinguish on panel c three areas that correspond to a specific actin filament organization, as will be seen later.
the droplet, resulting in contact areas of up to 6 mm diameter for a droplet volume of 10 µL. Because its persistence length is around 18 µm, a single filament can be considered to be confined when the film thickness is less than 5 µm. Using the wedge angle of 13°, it corresponds to a maximum distance of approximately 30 µm from the triple line. Starting from this point, the full evaporation of the solvent results in a remarkable and regular evolution of the droplet edge. The periodic interference fringes progressively vary near the contact line, and irregular structures deform the solution/air interface (Figure 1b). A rim is finally expanding from the droplet edge (Figure 1b and c). This growth phase corresponds to the accumulation of actin filaments, as will be seen later in fluorescence light microscope images. This process is transient only because the experiment shows that the droplet shrinks until the triple line detaches from this rim, as can be seen in Figure 2a-d. The triple line recedes with a typical microscopic contact angle of around 12°. Fugitive and diffuse interference fringes at the rear of the contact line can be interpreted as the drying of the remaining solute film. This thin wetting film can be observed on the graph of Figure 2 (between fringe 1 and 3 on segment AB), which reproduces the wedge profile
constructed from the interference fringes’ distribution observed with RICM. The receding of the triple line is unstable, and further pinning can be observed many times before complete drying of the droplet. 2. Collective Filaments Organization. To gain more insight into the drying process, the same phenomenon was observed in fluorescence microscopy mode. The phenomenological observations can then be related to the molecular and collective organization of the actin network resulting from the drying process. Figure 3 shows the fluorescent pattern remaining in the region of the initial pinning of the triple line once the solvent is completely evaporated. It is possible to distinguish three areas corresponding to different morphologies of the actin filaments’ organization. Some 20 µm from the initial triple line position (area 1) the filaments are well aligned parallel to the contact line, showing in some cases undulations perpendicular to the rim. In area 2, this organization is less pronounced with filaments frequently folded. This regime is characterized by a high fluorescence intensity suggesting that the film is thicker than in the first region. The evolution of the droplet edge and the deformation of the water/air interface observed by RICM (Figure 1a-c) are correlated with the accumulation of actin filaments in the droplet
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Figure 2. Detachment of the triple line of the partially wetting droplet from the rim of the droplet observed with RICM. The direction of the triple line displacement is indicated by the white arrow in panel c. The parallel stripes at the bottom of the images correspond to the original remaining rim formed at the beginning of the drying, as can be seen in Figure 1. The two plots correspond respectively to the intensity distribution (bottom) and the reconstructed wedge profile (top) of the section indicated by the double arrow on the intensity profile. In this case, the microscopic contact angle is about 12°.
wedge. The two former areas and corresponding actin structures are obtained during the intial pinning of the triple line. The third morphology of actin filaments remaining on the surface starts abruptly after the ordered domains and does not show any order (area 3). This randomly organized actin filaments film obtained during the receding of the triple line exhibits low fluorescence, suggesting that this layer is thinner than the rim formed by area 2. 3. Individual and Collective Filament Dynamics. Fluorescent actin filaments mixed with nonfluorescent actin filaments enable us to follow the single-filament dynamics associated with the three different actin film morphologies described previously. As already suggested above, the organization of actin filaments seems to be related to the dynamics of the triple line. Indeed, the organization is lost as soon as the triple line moves. Both casessthe pinned and the receding triple lineswill be considered in the following. After an initial spreading step, the triple line is pinned, and a flow of fluorescent actin filaments toward the front of the wedge is observed. Figure 4 shows that in the bulk area the filaments are aligned in the direction of the outward flow. There are only a few filaments that are not aligned, which is interpreted in terms of local entanglement. Note that at the actin concentration used in the present experiments the filaments would form an isotropic entangled network in the absence of external forces. This flow leads to an accumulation of filaments along the triple line. The filaments are stopped by the triple line or by the filaments that are already aligned. Two characteristic modes of behavior can be observed. Some filaments reorient and align parallel along the assemblies of aligned filaments (Figure 5). This behavior appears to occur during
the initial stages of the accumulation process near the triple line where the thickness of the droplet is typically less than 1 µm. This process of filament alignment can be seen in Figure 3 (area 1). The second mode of behavior consists of the buckling of filaments, which affects the whole length of the filaments (Figure 6) or their front part (Figure 7). It is noteworthy that the filament deformations appear to consist of a bending mode of defined wavelength (around 3 µm for the example shown in Figure 6). The buckling-like process occurs essentially when the wetting film is thicker than 1 µm, that is, if the confinement of the filament imposed by the wedge-like shape of the droplet is small, resulting in poorly organized filament assemblies as shown in Figure 3 (area 2). The flow of actin filaments toward the wedge stops when the triple line detaches from the rim and begin to recede. Under this condition, the translation of the center of mass of the filament is stopped, and the macroscopically motionless filaments adhere to the substrate while the solvent is retracting. This results in the deposition of disordered actin filaments at the rear of the receding contact line. Figure 8 shows the RICM image (right) and the corresponding fluorescence image (left) of the receding triple line. The retraction speed of the triple line is in this case 2.3 µm‚s-1 and exhibits a contact angle of 7°. We determined the velocity of the outward flux at the origin of the actin filaments’ accumulation in the wedge by tracking the filaments’ motion toward the wedge. The filament speed ranges from 3 to 13 µm‚s-1. To determine whether the motion of the filament reflected that of the solvent flow, we embedded fluorescent nanobeads into the actin solution. They exhibit a directed Brownian motion within the flowing solvent. The average speeds of
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Figure 3. Fluorescence micrograph of the film of actin filaments remaining after complete evaporation of the solvent. Three areas are selected and enlarged to show the three specific actin filament organizations in more detail. In area 1, the filaments show a nematic liquid-crystal-like alignment. In area 2, the filaments are still on the average parallel, but they are less well ordered. These two actin film morphologies result from the accumulation of the filaments as the triple line is pinned. In area 3, the filaments are completely random and do not show any organization. This morphology corresponds to the situation where the triple line is moving.
the beads in the direction of flow show a distribution similar to the speed of the filaments. For example, in a droplet where filaments flow with an average velocity of 8.9 µm‚s-1 the average bead velocity is 10.4 µm‚s-1 and ranges from 8.7 to 11.8 µm‚s-1. We therefore concluded that the average velocity of the filaments reflects the velocity of the fluid flow. Discussion The observation of an outward flow within an evaporating droplet is a common phenomenon that is described in the literature.11,40 This flow results from the combined effect of (1) the higher evaporation rate at the edge of the droplet and (2) the necessary replacement of the evaporated solvent in the region of the pinned triple line by solvent flowing from the center of the droplet. For this reason, a solute dispersed in an evaporating droplet with a pinned triple line will accumulate in the wedge (see Introduction for examples). In a same manner, the filaments are here dragged with the solvent toward the wedge of the droplet. It is noteworthy that the filaments are aligned in the flow. This configuration typical of polymer melts or solutions has been known for a long time on the basis of (40) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Phys. Rev. Lett. 2000, 62, 756.
indirect measurements such as birefringence and X-ray analysis. Such behavior is, however, rarely directly observed in the case of a single molecule or an assembly of molecules. Only a few observations of the alignment of DNA in a flow have been reported17,41-43 and theoretically investigated.44 Our observations differ from those mentioned above in two respects: (i) the semiflexible character of the actin filament (with the persistence length being on the order of the filament length) and (ii) the entanglement of the network of actin (whereas the DNA filaments are isolated). The flow mediates the accumulation of filaments in the wedge. The triple line and moreover the filaments already accumulated within the wedge act as a physical barrier for newly arriving actin filaments, whereas it is permeable to the solvent. Thus, each newly arriving filament is compressed against this barrier by the hydrodynamic flow of the solvent. The parallel assembly shows that the filaments are reoriented perpendicular to their orientation in the flow (i.e., parallel to the triple line). The process of (41) Jing, J.; Reed, J.; Huang, J.; Hu, X.; Clarke, V.; Edington, J.; Housman, D.; Anantharaman, T. S.; Huff, E. J.; Mishra, B.; Porter, B.; Shenker, A.; Wolfson, E.; Hiort, C.; Kantor, R.; Aston, C.; Schwartz, D. C. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 8046. (42) Smith, D. E.; Babcock, H. P.; Chu, S. Science 1999, 283, 1724. (43) Chopra, M.; Li, L.; Hu, H.; Burns, M. A.; Larson, R. G. J. Rheol. 2003, 47, 1111. (44) Buguin, A.; Brochard-Wyart, F. Macromolecules 1996, 29, 4937.
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Figure 4. Successive snapshots of fluorescent actin filaments flowing (parallel to the direction of the flow) toward the wedge of the droplet (0.5 s between consecutive snapshots). The gray area at the bottom of the pictures corresponds to the filaments accumulated at the rim of the droplet.
alignment is, however, the result of single-filament dynamics. We distinguished in the Results section two different dynamics, each leading to a more- or less-ordered filament packing. The most striking mechanism is the bending and folding of the filament. At first glance, this process, shown in Figure 6, seems to be equivalent to a buckling process driven by the mechanical instability of a rigid rod upon compression. Following the Euler theory of buckling, the instability can be characterized in terms of a characteristic bending length lb.45 This bending length is the contour length of the filament from the tip, where it touches the compressed actin barrier, to the first curvature maximum of the contour. In our case, the bending instability is caused by hydrodynamic stress parallel to the rod axis. The problem is equivalent to the situation of a rigid rod buckling under its own weight in a gravitational field (45) Landau, L. D.; Lifschitz, E. M. Theory of Elasticity, 3rd ed.; Pergamon Press: Oxford, U.K., 1986.
because every section along the filament is subjected to the same hydrodynamic shear force. The drag force per unit length can be expressed in terms of the length and diameter of the filament and the solvent velocity, which are all known parameters. For a given length and bending rigidity, the theory of mechanical instability yields the first deflection maximum lb at which the filament will be folded and will finally break. If one compares this calculated value of lb with the value obtained from the reconstructed filament contours (shown in Figure 6), it is found that these values always differ by at least a factor of 3. Therefore, a classical description of the buckling instability process seems to be an inappropriate model to describe the bending and folding of an actin filament compressed against a barrier, as observed in our case. In the following, we propose that this discrepancy can be explained by taking into account the dynamic properties of actin filaments. In contrast to a rigid rod that is buckling from a rigid state, actin filaments exhibit thermally excited bending undulations. Therefore, these filaments also
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Figure 5. Successive snapshots of an actin filament extended over its whole length (0.5 s between consecutive snapshots) in the area corresponding to area 2 in Figure 3. The red line indicates the average profile of the filament flowing toward the triple line.
exhibit nonzero curvature in the absence of an external load. In addition, both ends can move freely in the direction of the load, and the translational motion is restricted only by the network. The motion can thus be described in terms of the reptation model. Moreover, the length of a filament (10 to 30 µm) and its persistence length (∼18 µm) are large compared to the thickness of the film (around 3 µm) in which the bending is observed. This allows us to apply in the following the tube model of polymer self-diffusion. Because of the restriction of the filament motion by the walls of the tube, the characteristic length lb is expected to be determined by the tube diameter. Therefore, we propose here that the dynamic instability of the filament can be considered to be an amplification of an excited undulation mode by the tangential load. The tube diameter is determined by the mesh size (530 nm for a 10 µM actin solution).46 Undulation modes with amplitudes exceeding the tube diameter will be suppressed.21,47 For the case shown in Figure 6, the lowest mode following this condition is the fourth mode of wave vector q ) 4π/L exhibiting an amplitude of 529 nm and a characteristic wavelength lb ) 4.45 µm. This value has to be compared with the observed characteristic length lb ) 3 µm that lies between the value corresponding to the sixth and seventh modes, which would correspond to tube diameters of 240 and 168 nm, respectively. A likely explanation for this discrepancy is the local increase of the actin concentration near the droplet edge due to the drying process. A second essential parameter determining the amplification of a distinct mode is the relaxation time τr. It is expected that the mode of the relaxation time that coincides with the characteristic time τc of the hydrodynamic excitation process will be preferentially amplified. (46) Isambert, H.; Maggs, A. C. Macromolecules 1996, 29, 1036. (47) DeGennes, P. G. Scaling Concepts in Polymer Physics, 2nd ed.; Cornell University Press: Ithaca, NY, 1985.
Figure 6. Successive snapshots of an actin filament buckling over its whole length with a characteristic length of around 3 µm when crashing against filaments already accumulated in the wedge of the droplet (0.5 s between consecutive snapshots) in the area corresponding to area 2 in Figure 3. The red line indicates the average profile of the filament flowing toward the triple line.
The characteristic relaxation time τr determines the time required for the excitation of specific modes.48 The actin filament shown in Figure 6 exhibits a characteristic bending mode of wavelength 3 µm, which corresponds roughly to the sixth or seventh undulation mode. The relaxation times are 85 and 48 ms, respectively. To determine the characteristic time τc of the hydrodynamic excitation process leading to the bending of the actin filament, we determined the time dependency of the bending energy per unit length Eb/l for each contour y(x) of the filament (stored in the movie). The bending energy is calculated using the following equation
Eb )
( ) 2
∂y ∫ol ∂x 2
κ 2
2
dx
κ is the bending rigidity of a rhodamine-phalloidinstabilized actin filament, which is roughly 6 × 10-26 J‚ m.49 (48) Wiggins, C. H.; Riveline, D.; Ott, A.; Goldstein, R. E. Biophys. J. 1998, 74, 1043.
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Figure 7. Successive snapshots of an actin filament buckling partially (0.5 s between consecutive snapshots) in the area corresponding to area 2 in Figure 3. The red line indicates the average profile of the filament flowing toward the triple line.
Figure 9 shows the calculated values for the contour length density of the bending energy Eb/l plotted versus time for a frame rate of 62 ms. It appears from the snapshots that the bending is associated with a strong increase in the bending energy per unit length. The bending energy can be fit with an exponential function of type
Eb(t) Eb(0) ) (1 + et - t0/τc) l l The fit parameters are the equilibrium bending energy per length Eb(0), the characteristic time τr for the increase in the bending energy, and the time t0 at which the increase starts. The retrieved value of t0 is 0.6567 ( 0.0054 s and coincides well with the observed time at which the filament encounters the barrier. The value obtained by the above fitting procedure for the equilibrium bending energy per unit length of (1.77 ( 0.21) × 10-16 J/m corresponds to a total bending energy of (3.72 ( 0.44) × 10-21 J of the entire filament, which coincides very well with the expected value of the thermal energy kBT of ∼4 × 10-21 J. The charac(49) Roos, W. H.; Roth, A.; Konle, J.; Presting, H.; Sackmann, E.; Spatz, J. P. ChemPhysChem 2003, 4, 908.
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teristic time for the increase in the bending energy was determined to be 72.9 ( 5.4 ms, which is in the range between the two boundary values of 85 and 48 ms expected for the characteristic bending length of 3 µm. The same analysis was carried out for other filaments, which were observed to bend and fold, when they encountered the dense actin barrier at the rim of the droplet. For all cases studied, the above result was verified. This provides very strong evidence that the bending length of the filaments is determined by the response time of the bending process. It is difficult to estimate the conditions under which nematic ordering or buckling of the filaments is observed. The nematic ordering is a consequence of the confinement of the filaments transported toward the rim of the droplet and is thus preferentially observed close to the rim of the droplets. This interpretation is also supported by the fact that buckling is more frequently observed at distances of a few tens of micrometers from the rim where the filaments are less constrained. Another explanation is the reduced actin concentration at the rim during the initial phase of spreading. The high affinity between the aqueous phase and the hydrophilic surface can lead to fast spreading of the solution with only a few filaments dragged toward the front. A lower actin concentration would allow the filaments approaching the rim to reorient more freely than the filaments in the bulk. Unfortunately, it is difficult to estimate the local concentration of the actin solution and to relate it to the length of the well-organized zone (area 1 in Figure 3). Our results show that the pinning of the triple line is one of the conditions for the organization of the filaments because it ensures the flow at the origin of the filaments’ accumulation. The pinning is, however, only transient because the triple line retracts again. The solute evaporation leads to a decrease in the volume of the droplet for a constant area of contact with the substrate. This increases the unbalanced Young force until it is large enough to induce the retirement of the triple line. In this phase, the flow process is impeded in the droplet because the shrinking of the droplet is compensated for uniformly and instantaneously over the whole liquid/air interface of the evaporated volume. The immobile actin network is then fixed to the substrate because of the fast retirement of solvent from the receding of the triple line (Figure 8). This process finally results in an unorganized actin film (area 3 in Figure 3) at the rear of the receding triple line because the mobility becomes too low to enable the reorientation of the filaments. Our experiments suggest a dynamic isotropic-nematic transition50,51 at some radial distance from the pinning line. It is attributed to the interplay of directed convective flow and local crowding of filaments near the rim of the droplet. (While near the center, the concentration is small enough to enable isotropic entanglement.) The nematic ordering can be mediated by two mechanisms: first by the crowding of the filaments near the rim due to the hydrodynamic flow and solvent evaporation and second by the hydrodynamic shear forces that are expected to rotate or fold the filaments when they reach the rim. The theory of the isotropic-nematic transition was proposed by Onsager. According to Furukawa et al.,50 the concentration of actin corresponding to the transition is given by CL2d > 4.25, where C is the filament concentration, L is the filament length, and d is the filament diameter. If one assumes a monodisperse length distribution of the fila(50) Furukawa, R.; Kundra, R.; Fechheimer, M. Biochemistry 1993, 32, 12346. (51) Coppin, C. M.; Leavis, P. C. Biophys. J. 1992, 63, 794.
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Figure 8. Comparison between fluorescence (left) and RICM (right) images of the same spot of a droplet wedge showing the situation of a receding triple line. The image on the left shows an actin filament film formed at the rear of the receding triple that does not show any order. The white strokes indicate the direction of the triple line displacement.
between the actin cortex and the cell cytoplasm. It is generally assumed that this liquid crystal formation is induced by cross linkers such as filamin, R-actinin, or bivalent ions.52 This work suggests an alternative route for the rapid self-assembly of parallel arrangements of the freshly polymerized actin filaments in cells, which is expected to be much faster than the generation of parallel actin assemblies by a cross linker during polymerization. In cells, strong forces on the order of 100 pN generated by still unknown mechanisms induce local flows over distances of a few micrometers. The associated hydrodynamic flow together with the crowding of the actin by de novo polymerization enforces the parallel organization of the freshly formed filaments. The cross linking of these actin bundles into stress fibers could occur as a second step. Figure 9. Variation of the bending energy Eb/l with time corresponding to the deformation of the filament shown in Figure 6 (frame rate of 62 ms).
ments (20 µm) and a diameter d ) 7 nm, then one expects an isotropic-to-nematic transition at a concentration of 18 µM, which is close to our experimental conditions (between 2.5 and 20 µM). The experiments were performed for long filaments with length on the order of 20 µm. The ordering we observed could play a role in cells where cell stimulation often causes the rapid formation of stress fibers by the rapid increase (time scale of 1 s) of intracellular content of the polymerized fraction of actin. This process occurs near the interface
Acknowledgment. The work was supported by the Deutsche Forschungsgemeinschaft (SFB 413) and the Fonds der Chemicher Industrie. L.V. acknowledges the fellowship received from the Centre de Coope´ration Universitaire Franco-Bavarois CCUFB-BFHZ (http:// www.bfhz.uni-muenchen.de/), and L.L. acknowledges the fellowship received from the Alexander von Humboldt Foundation. We also kindly thank Professor Erwin Frey and Dr. Hamidou Haidara for fruitful discussions. LA050986W (52) Borukhov, I.; Bruinsma, R. F. Phys. Rev. Lett. 2001, 87, 158101.
