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Laboratoire Physico-Chimie Curie, UMR 168, Institut Curie/Centre National de la Recherche Scientifique/University Paris 6th, Paris, France
Correspondence: Address reprint requests to C. Sykes, Tel.: 33-1-42-34-67-90; E-mail: cecile.sykes{at}curie.fr.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUDING REMARKSAPPENDIX A: DERIVATION OF...APPENDIX B: VELOCITY FIELD...SUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUDING REMARKSAPPENDIX A: DERIVATION OF...APPENDIX B: VELOCITY FIELD...SUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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). Importantly, cross-linkers are necessary for efficient force generation by actin networks: cells depleted in the cross-linking protein filamin, for example, do not form lamellipodia, but produce spherical bare membrane protrusions called blebs (2). The importance of actin cross-linkers suggests that the elasticity of the actin network plays an important role in force generation within cells.
Actin polymerization can also propel bacteria such as Listeria monocytogenes and endosomes inside cells (3–5). In simplified systems, bacteria, beads, liposomes, and oil droplets can likewise move in cell extracts or in a medium containing a minimum set of purified proteins (6–11). The cargos are covered with nucleation factors, which activate actin polymerization at their surface. Actin filaments then grow and form a cross-linked gel around the object. In many cases, the actin shell is polarized from the beginning: bacteria and liposomes are often asymmetrical, polarity being imposed during bacterial division or during liposome formation (3,5,9). However, even for symmetrical objects, such as uniformly-coated beads, the gel can spontaneously break, leading to the formation of an actin comet that propels the object forward (12).
Beads propelled by actin polymerization have been widely used as a model system for Arp2/3-dependent actin-based movement. The molecular basis of this movement is now well understood (13). Once activated at the bead surface by a Wiskott-Aldrich Syndrome Protein, the Arp2/3 complex nucleates new filaments as branches on preexisting filaments, thus leading to the formation of a dendritic actin network. Capping protein or gelsolin limits the growth of actin filaments to a region close to the bead surface, while actin depolymerizing factor (ADF)/cofilin regulates filament depolymerization at the minus end of filaments. Together with profilin, these proteins cooperate to ensure a high level of ATP-coupled actin monomers for further polymerization (14). Cross-linking proteins modulate the physical properties of the actin network and the time necessary for comet formation increases with their concentration (12). The interplay of all these ingredients ensures the continuous growth of the actin network from the bead surface providing the forces needed for movement.
Various physical models have been proposed to explain how actin polymerization can generate forces (15–17). It is now well documented that the elastic properties of actin gels are responsible for symmetry breaking (12,18). The role of elasticity for bead movement, however, is still under discussion, although several observations suggest that the comet exerts elastic stresses on the bead during its movement (8–11). Here, we use a simple assay where beads coated with verprolin/cofilin/acidic domain (VCA), an Arp2/3 activator derived from Wiskott-Aldrich Syndrome Protein, move in a mixture of commercially available purified proteins, and we study the influence of cross-linkers and of bead size on the velocity. By photobleaching fluorescent actin to mark regions in the gel, or by using successively two different actin markers, we also provide direct evidence that the actin gel in the comet undergoes deformations that relax as the comet grows. These observations can be accounted for by a model in which the elastic properties of the actin network and diffusion of actin monomers result in strains in the comet that relax as the comet grows.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUDING REMARKSAPPENDIX A: DERIVATION OF...APPENDIX B: VELOCITY FIELD...SUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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-actinin, VCA, and biotinylated actin were purchased from Cytoskeleton (Denver, CO), and used without further purification. Protein concentrations were determined by SDS-PAGE using a BSA standard. Alexa Fluor 594-labeled actin (red actin) and Alexa Fluor 488-labeled actin (green actin) were from Molecular Probes (Eugene, OR) and streptavidin from PerBio Science (Brebières, France). Fascin was purified as described in Vignjevic et al. (19). Human filamin A was a gift of F. Nakamura and T. Stossel (Brigham and Women's Hospital, Boston, MA).
Bead preparation and motility assay
Polystyrene beads (Polysciences, Warrington, PA) of different radii were coated with VCA (at complete saturation) as described previously (12), and stored in a storage buffer (10 mM HEPES pH 7.5, 0.1 M KCl, 1 mM MgCl2, 0.1 mM CaCl2, 1 mg/ml BSA) for up to one week. F-actin was prepared by adding 0.2 mM EGTA, 0.1 M KCl, and 1 mM MgCl2 to a solution of G-actin. Unless otherwise stated, the motility medium contained 10 mM HEPES pH 7.5, 0.1 M KCl, 1 mM MgCl2, 0.15 mM CaCl2, 1.8 mM Mg-ATP, 6 mM DTT, 0.13 mM Dabco (an anti-photobleaching agent), 8.1 µM F-actin (10% labeled with Alexa Fluor 594 or Alexa Fluor 488), 0.1 µM Arp2/3, 0.35 µM gelsolin, 3 µM ADF/cofilin, 1 µM profilin, and 10 mg/ml BSA. Samples were prepared by diluting a small volume of bead suspension 30 times in the motility medium, and placed between a glass slide and coverslip (18 x 18 mm) sealed with vaseline/lanolin/paraffin (1:1:1). The total sample volume was such that the spacing between slide and coverslip was at least three times the bead diameter. The comet length was measured as a function of time for 8–12 different beads. Since no significant depolymerization was observed during the experiment for the ADF/cofilin concentration used, the average bead velocity could be estimated as the average rate of increase of the comet length.
Bead observation and data processing
Phase contrast and fluorescence microscopy were performed using an Olympus inverted microscope with a 100x oil immersion objective (Olympus, Rungis, France). Confocal microscopy experiments were carried out with a Zeiss confocal microscope with a 63x 1.4 NA oil immersion Plan-Apochromat objective, and controlled by LSM 510 META software (Carl Zeiss, Jena, Germany). Actin-Alexa Fluor 488 was observed with an Ion Argon 25mW laser (488 nm) at 0.3–2% of the laser power. For photobleaching, 100% of the laser power was used. Image contrast was reprocessed using MetaMorph software (Universal Imaging, Downington, PA).
Determination of G-actin concentration in the medium
The motility medium, in the presence or absence of cross-linkers, was left at room temperature for 20 min and centrifuged at 150,000 x g during 45 min to sediment F-actin. The actin concentration in the supernatant (G-actin) was determined by Western blot using a monoclonal anti-actin, mouse IgG antibody (Sigma-Aldrich, St. Louis, MO).
Viscosity measurements
Viscosity measurements were performed by immersing BSA-coated beads of different radii (R = 1–4 µm) in the motility medium and observing their Brownian motion using bright-field microscopy. The mean-square displacement of the beads, 
r2
, increased linearly as a function of time: r2 = 6Dt, where D is the diffusion coefficient of the beads. The viscosity 
was calculated from the slope of this line using the Stokes-Einstein relation: D = kT/6
R.
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RESULTS |
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The effect of the bead radius on the velocity depends on the concentration of gelsolin in the medium (Fig. 1 a). At 0.35 µM gelsolin, the velocity decreases approximately as 1/R, whereas at higher gelsolin concentrations the effect of the radius on the velocity is much smaller. For 0.73 µM gelsolin, the velocity is almost constant for radii smaller than 2.5 µm, and it is only 
25% lower for 5-µm radius beads.
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-actinin and fascin only a decrease in velocity could be observed (Fig. 1 c). We checked that the effect of cross-linkers was not due to a change in the bulk properties of the medium affecting either the availability of G-actin monomers or the viscosity of the solution. The concentration of G-actin in the bulk in the presence and in the absence of cross-linkers was measured by centrifuging the samples and determining the G-actin concentration in the supernatant using Western blot analysis. No significant change in the monomer concentration was observed after the addition of cross-linkers (within the measurement error of ±50%, see Supplementary Material Fig. S2). The effect of cross-linkers on the viscosity was measured by monitoring the Brownian motion of suspended BSA-coated beads. In the mixture of purified proteins but in the absence of streptavidin the measured diffusion coefficient is approximately three times smaller than that of beads in the buffer, indicating that the motility medium is three times more viscous than the buffer. Surprisingly, after the addition of streptavidin, the sample viscosity was found to decrease and was almost equal to that of a buffer at 0.8 µM streptavidin (see Supplementary Material Fig. S3).
Evolution of a newly-grown actin layer
Elastic stresses that build up in the actin gel are responsible for symmetry breaking and comet formation (12). To determine if stresses are still present after symmetry breaking in a growing comet, we monitored the evolution of a newly-grown layer of actin at a stage when the comet was already formed: we used successively two different actin markers during the comet growth. Our first attempt was to add actin of a different color to a mixture containing beads with comets. This experimental way was not satisfactory, since actin comets appeared inhomogeneous by phase contrast microscopy, probably due to a sudden increase in total actin concentration. We decided to use a different experimental approach that would preserve the homogeneity of the comet. Beads placed in a medium containing red actin (10% of total actin) and beads placed in a medium containing green actin (10%) were incubated separately at room temperature. After 45 min, the two preparations were added together and gently mixed. No change of the structure of the comet could be seen in phase contrast using this approach (Fig. 2 a). The evolution of a newly inserted actin layer was monitored by observing the interface between red and green actin in the comet using fluorescence microscopy (Fig. 2). Our observations clearly show that the red/green actin interface undergoes deformations. The radius of curvature of this interface is initially that of the bead, and then increases as the actin grows from the bead surface. This effect is more pronounced near the sides of the comet (Fig. 2 a). The red-green interface stops changing shape when the layer is further than approximately one-bead's diameter from the bead surface (see Supplementary Material Movie 1). These results provide evidence for deformations in the actin comet during bead motion. No significant difference was observed in the shapes of the red-green interfaces at 0.2, 0.35, or 0.92 µM gelsolin (Fig. 2, b and c).
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15° as the comet evolves (Fig. 3 a). |
DISCUSSION |
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,21). As a result, deformations in the gel result in elastic stresses that may affect the movement of the beads. Although elastic stresses have been shown to be important for symmetry breaking (12), for squeezing of vesicles or oil droplets by actin tails (9–11), and for the hopping motion of beads (8), their role in motility and their effect on speed remain under discussion. The deformations in the gel (Figs. 2 and 3) and the effect of cross-linkers (Fig. 1, b and c) provide support for a role of elasticity in actin-based propulsion.
Elastic model for bead movement
Polymerization at the bead surface pushes away older gel layers, resulting in an accumulation of elastic stress due to bead curvature (22). Relaxation of this elastic stress as the bead moves forward results in an effective propulsive force Fel Eh3/R (23), where E is the elastic modulus of the gel, h the thickness of the gel on the side of the bead, and R the bead radius (see Fig. 4 b). The propulsive force is opposed by a friction force Ffr 
vR2, where is a friction coefficient per unit area, which can be related to the kinetics of attachment and detachment of filaments (24). The bead velocity v follows from the balance between elastic and friction forces.
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multiplied by the time R/v spent by the gel at the surface of the bead: 
(20,24). This gives the velocity
 | (1) |
For higher elastic moduli, the accumulated elastic stress may limit actin polymerization at the surface. In this case, the gel thickness saturates at a value close to the stress-limited steady-state thickness of the gel before symmetry breaking (22): 
, with 
µ the chemical energy of the polymerization reaction, and 
0 a characteristic stress for the variation of the polymerization velocity with stress. The characteristic stress 0 is related to the mesh size 
of the actin network, and the size a of an actin monomer by 
. In this limit, the velocity is approximately
 | (2) |
The transition between these two regimes occurs when the modulus E reaches the value for which the velocities given by Eqs. 1 and 2 are equal and will be discussed further below.
Velocity is limited by diffusion and by elasticity at low and high gelsolin concentrations, respectively
In both the polymerization-limited (low E) and stress-limited (high E) regimes, the elastic propulsion model predicts that the velocity does not depend on the bead radius. This is indeed what we observe at high gelsolin concentrations: at 0.73 µM gelsolin, the velocity is independent of the radius for radii smaller than 2.5 µm (Fig. 1 a), in agreement with previous observations (25). However, at lower gelsolin concentrations, we observe a decrease in the velocity with increasing radius (0.35 µM and 0.54 µM gelsolin in Fig. 1 a). To account for this R-dependency, we must take into account the fact that monomers consumed during polymerization have to diffuse to the bead surface and that this can significantly reduce the actin monomer concentration inside the comet if the actin meshwork is dense enough (26,27). The diffusive flux of monomers toward the bead surface at the center of the comet can be approximated as 
, with D the diffusion coefficient of monomers in the gel and C
and CS the monomer concentrations at the exterior surface of the comet and at the bead surface at the center of the comet, respectively (see Fig. 4 b). We assume that monomer diffusion in the bulk of the solution is fast, so that the actin monomer concentration outside the gel is approximately equal to the concentration in the bulk. At steady state, the diffusive flux is equal to the monomer consumption per unit area, JP = kCS/2 with k the polymerization rate constant (which depends on the local stress at the rear of the bead), and the distance between growing filaments (of the order of the mesh size of the gel). The steady-state monomer concentration is then
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2. At the center of the comet, the polymerization velocity vp = kCSa must equal the velocity of the bead, so that in the diffusion-limited regime the speed is
 | (4) |
Our results for 0.35 µM gelsolin (Fig. 1 a) can be fitted by Eq. 4, giving CD2a 
0.25µm2/min. With D = 2µm2/s (26), C = 1 µM (measured by Western blot, see Supplementary Material Fig. S2), and a = 2.7 nm, the fitted value gives = 36 nm, in agreement with estimates from electron micrographs (26). Under these conditions, taking k = 12 µm–1 s–1 = 0.02 µm3 s–1 (28), the dimensionless parameter D2/kR varies between 0.13 and 0.026 for radii between 1 and 5 µm, which is consistent with our assumption of diffusion-limited kinetics (D << kR/2).
Diffusion versus elasticity
The curves shown in Fig. 1 a are well accounted for by a diffusion-limited regime at low gelsolin concentration and an elasticity-dominated regime at high gelsolin concentration. This effect of gelsolin can be explained by its capping activity. At higher gelsolin concentrations, the number of growing actin filaments is smaller due to increased capping, therefore the mesh size is larger. This is supported by the observation that the gel is less dense for higher gelsolin concentrations, as seen from the fluorescence intensity of the comet (see Supplementary Material Fig. S4). Above a critical mesh size 
, elasticity becomes dominant. Assuming that D and k do not vary significantly, we find c = 100 nm for R = 1 µm and c = 224 nm for R = 5 µm, which is in the experimentally accessible range (22,26). Note that the diffusion coefficient D could also increase with the mesh size, which would reduce c.
The above discussion shows the existence of three distinct regimes for bead movement: a diffusion-limited regime (velocity given by Eq. 4) and two elastic regimes where diffusion is not rate-limiting, corresponding to low gel elastic moduli where the thickness is limited by polymerization (Eq. 1) and high elastic moduli where the thickness is limited by the elastic stresses (Eq. 2). The actual state of the system is determined by eight variables that can be recast into two reduced variables: a dimensionless modulus 
and a dimensionless diffusion coefficient 
, where is the actin polymerization velocity in the absence of any external stress when the monomer concentration is C. The details of the derivation of e and d are given in Appendix A. The e–d plane is divided into three different regions corresponding to the three regimes discussed above. As explained in Appendix A, the transition between the polymerization-limited regime and the stress-limited regime occurs at e = 1. If e < 1, the crossover between the diffusion-limited regime and the polymerization-limited regime occurs when e = d 4, and when e > 1, the transition between the stress-limited regime and the diffusion-limited regime occurs when e = d–2. Any experiment where one of the parameters e or d is varied can be considered as a curve in this diagram along which the velocity varies. Note that varying the bead radius changes only the coordinate d, and that it corresponds to a horizontal line in the diagram.
Velocity is affected by the elastic modulus
We observe that the presence of cross-linkers changes the speed of the beads (Fig. 1, b and c). This effect could result from depletion of available actin monomers, since cross-linking might stabilize actin filaments and thus prevent depolymerization in the bulk solution (29). However, our measurements of the G-actin concentration in the samples show that the presence of cross-linkers does not change G-actin concentration within the experimental error (see Supplementary Material Fig. S2). Moreover, a decrease in monomer concentration would imply a decrease in velocity and could not account for the velocity increase observed at high gelsolin and cross-linker concentration. Nor can the velocity variations be accounted for by a change in the medium viscosity. Indeed, the medium viscosity is at most three times higher than that of the buffer and it decreases by roughly a factor of 3 after addition of cross-linkers (see Supplementary Material Fig. S3), probably because the medium becomes inhomogeneous due to local bundling of actin filaments leading to a lower F-actin concentration elsewhere. Given that the force generated by the growing comet is in the nanoNewton range (20), and that the viscous drag force is of the order of 10 femtoNewtons (16), we conclude that the viscous force is negligible compared to the polymerization force in our experiments. It therefore seems likely that the effects of cross-linkers on bead velocity are due to a change in the elastic properties of the actin comet.
At 0.35 µM gelsolin, the velocity decreases strongly after addition of cross-linkers for all four cross-linkers (Fig. 1, b and c). As indicated by the radius dependence shown in Fig. 1 a, the speed is limited by diffusion at this gelsolin concentration when no cross-linkers are present. If the system stays in the diffusion-limited regime, the strong decrease in velocity suggests a decrease in the diffusion coefficient when cross-linkers are added. This effect might be a consequence of a decrease in the mesh size upon addition of cross-linkers, thus preventing diffusion of monomers through the gel. Alternatively, the decrease in velocity can be explained by a crossover to the stress-limited regime (Fig. 4 a): assuming that the addition of cross-linkers mainly increases the elastic modulus E, we move in the positive e-direction in the state-diagram upon addition of cross-linkers. At e = d–2, there is a crossover from the diffusion-limited to the stress-limited regime. After the crossover, the velocity decreases with the elastic modulus as v E–1/2 (Eq. 2).
At higher gelsolin concentration, diffusion is not rate-limiting (see above), and the effect of cross-linkers is different than at low gelsolin concentration. Upon addition of streptavidin and filamin, the bead velocity first decreases and then strongly increases (Fig. 1, b and c). The increase in the velocity at high cross-linker concentrations can be explained by an increased modulus (Eq. 1), and provides strong support for the elastic propulsion mechanism where one expects v to vary as E1/4 when the gel thickness is determined by the polymerization rate. However, the comparison between experiments and the predicted scaling laws can only remain qualitative since the exact relation between the modulus E, the friction coefficient , and the concentration of cross-linkers is not known. Nevertheless, the increase in velocity upon addition of cross-linkers indicates that we are in the polymerization-limited regime (e < 1) at high gelsolin concentration, probably because the friction coefficient is small due to a low number of attached filaments on the surface. It remains unclear as to why the velocity decreases at lower amounts of cross-linkers, but this might be due to an increase of the friction coefficient upon filament bundling close to the surface (Eq. 1). One could expect that for very high cross-linker concentration (and thus modulus E), the system would reach e = 1 and crossover to the stress-limited regime, leading to a decreasing velocity (Fig. 4 a). We have not observed such a decrease in our experiments, however. The reason why e does not increase sufficiently to reach the transition might be that e varies proportionally to E4, so that an increase in E upon addition of cross-linkers is counterbalanced by a decrease in .
For -actinin and fascin, we do not observe an enhancement of the movement, probably because their effects on the modulus and the stresses in the gel are smaller. Note that the effect of these bundling proteins for symmetry breaking was also much smaller than that of biotin/streptavidin or filamin (12).
Effect of elasticity and diffusion on deformations in the gel
The observation of a newly-grown actin layer in the two-color experiment shows that deformations in the comet occur at low as well as at high gelsolin concentrations (Fig. 2). Indeed, both diffusion and elastic properties of the comet can account for the deformations observed.
In the diffusion-limited regime, polymerization at the center of the comet is slow due to a reduced supply of monomers, while polymerization at the sides of the comet, which can be easily reached by monomers, is faster. As a result, in the reference frame of the bead, the gel moves faster in the outer part of the comet than in the central part. This accounts for the deformations of bleached vertical lines that deform as whiskers (Fig. 3) as well as for the observation that new-grown actin layers open up and reduce their curvature (Fig. 2).
In the elasticity-dominated regime, as actin polymerizes at the bead surface in a perpendicular direction, it pushes away older gel layers, resulting in an accumulation of elastic stress due to the bead curvature (22). A newly-grown gel layer is thus first stretched because new material is continuously added at the bead surface leading to stresses in the layer. As the layer gets further away from the bead surface, it can reduce these stresses by reducing its area. Given the geometry of the comet, a decrease in area is achieved by reducing the layer curvature (Fig. 4 b). This is what we observe in the two-color experiments (Fig. 2), and is also supported by the photobleaching experiments, where a perpendicular line turns into a curved line oriented with its concavity toward the comet extremity (Fig. 3). In Appendix B, we propose a more detailed elastic model for a thin layer of gel on the bead surface that accounts for the deformation of lines bleached when the gel is still around the bead.
In most cases, probably both diffusion and elasticity contribute to the deformations observed. Importantly, both mechanisms result in strains and therefore elastic stresses in the actin gel. The stresses relax as the comet grows and no deformation is observed further than approximately one-bead's radius from the bead surface (see Fig. 2 b, Fig. 3, and Supplementary Material Movie 1).
Actin-based motility within cells
Listeria monocytogenes or endosomes move within cells by activating actin polymerization at their surface. Since a cell is not a homogeneous medium and is constantly remodeling its cytoskeleton, it is essential to understand the behavior of such intracellular motile objects under different conditions. The bead system is a powerful tool for this, because it allows for a systematic variation of parameters like the bead size or the presence of various actin-binding proteins. From the present work, we know that objects of different sizes move at the same speed if the diffusion coefficient of actin monomers is sufficiently high (see Eq. 4). Under such conditions, the speed will only be affected in cellular regions where actin-binding proteins like cross-linkers or branching agents are effective. On the other hand, if actin monomer diffusion through the comet is impeded for example by transient interactions with the comet actin network, large objects will move more slowly than small objects. This effect of actin monomer diffusion might explain why two different experiments give apparently contradictory results on the effect of size on speed (7,25).
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CONCLUDING REMARKS |
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APPENDIX A: DERIVATION OF THE STATE DIAGRAM |
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The crossover from the elastic polymerization-limited regime (corresponding to small elastic moduli E) to the elastic stress-limited regime (high E) occurs when the elastic modulus reaches the value for which the velocities given by Eqs. 1 and 2 are equal:
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Introducing a dimensionless elastic modulus e,
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The crossover between the elastic polymerization-limited regime and the diffusion-limited regime occurs when the velocities given by Eqs. 1 and 4 are equal,
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 | (6) |
We then introduce a dimensionless diffusion coefficient d:
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Using e and d, Eq. 6 simply reads e = d 4.
Finally, the crossover between the elastic stress-limited regime and the diffusion-limited regime occurs when the velocities given by Eqs. 2 and 4 are equal,
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These relations determine the boundaries between the three regimes of movement. When e < 1 and e < d 4, the system is in the elastic polymerization-limited regime. When e > 1 and e > d –2, the system is in the elastic stress-limited regime. Otherwise, the system is in the diffusion-limited regime. Thus, the values of the dimensionless coefficients e and d fully determine the regime in which the system resides. These results are summarized in a state diagram in the (e–d) plane, where each couple of coordinates (e, d) corresponds to a set of experimental conditions (Fig. 4 a).
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APPENDIX B: VELOCITY FIELD IN A THIN LAYER OF GEL AT THE BEAD SURFACE |
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The constitutive equations of the elastic material are given by 
, where ij is the deviatory stress tensor related to strains of the fluid elements, and eij = 1/2(
jvi + ivj) is the velocity-gradient tensor (30,31) (in the convective derivative D/Dt, we ignore the rotational contribution associated to the vorticity that turns out small). For a two-dimensional gel, only the components exx, exy = eyx, and eyy of the velocity-gradient tensor do not vanish. Then, in a steady state, the constitutive equations give three relations:
 | (8) |
 | (9) |
 | (10) |
The total stress tensor is given by ij – p 
ij where p is the pressure, and the local force balance equation reads as
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 | (12) |
Finally, we consider the gel as an incompressible fluid so that 
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On the bead surface, we suppose that there is viscous friction with a friction coefficient per unit area and that the polymerized material is not under tension so that, at y = 0, xx – p = 0, and xy = vx. At the outer surface of the gel, the normal component of the stress tensor vanishes. In a thin film approximation, the angle between the free surface and the x axis, dh/dx, is small (Fig. 5 b), and we have the following limit conditions at y = h(x):
 | (13) |
 | (14) |
Since lines along the x axis do not deform during comet growth (Fig. 3 c), the component vx of the velocity mainly accounts for the observed deformations. As h << L, we expand all quantities in powers of y. This approximation is supported by the experimental observation that a line along the y axis bleached at the surface of the bead, remains straight during comet growth: an expansion of vx up to first-order in y is therefore sufficient to account for the deformation (Fig. 3 b). We write vx = v0 + ß(x)y and xx – p = s(x)y.
The incompressibility condition gives the perpendicular velocity 
, and the actin mass conservation imposes a relation between the velocity and the local thickness: v0(x)h(x) = vpx, where we have chosen the origin of the coordinate x at the contact line between the comet and the bead. The transverse stress xy can be determined from Eq. 11: 
.
A first relation between the velocity v0 and the normal stress coefficient s is then obtained from the boundary condition for the normal stress equation, Eq. 13:
 | (15) |
A second relation between s(x) and v0 is obtained by first calculating the normal stress 
from Eqs. 12 and 14 and then using the constitutive Eqs. 8 and 10:
 | (16) |
Combining these two relations, we derive a differential equation for the velocity v0(x):
 | (17) |
This equation can be solved by introducing the variable 
. In the limit where V >> 1, the solution is
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the contact angle between the comet surface and the x axis (the bead) at the front of the gel (as shown in Fig. 5 b), and a the size of a monomer.
Dimensional analysis of Eq. 17 gives 
, and 
(actin flux conservation). If the angle is not small (which seems true experimentally; the theoretical derivation of would require a detailed description of the crossover to the cylindrical comet behind the bead, which is beyond the scope of this work), V2(a) >> 1. Since x/a is at most of the order of 1000 (a = 2.7 nm and L R 2.5 µm), we thus have V2(x) >> 1, which is consistent with our assumptions. Note that the scaling law that we obtain is slightly different form that obtained in the elastic scaling model presented in the text. This is probably due to the thin film approximation used here.
 | (18) |
The coefficient ß that gives the variation of the velocity in the y direction perpendicular to the bead is directly deduced from v0 by writing the constitutive Eq. 9 for y = 0:
 | (19) |
We thus find that ß is positive, which is in agreement with experimental observations. Indeed, vx = v0 + ßy, with 
> 0, implies that the gel in the external part of the comet moves with a higher velocity than the gel close to the bead surface, leading to deformations in the same direction as those observed experimentally (Fig. 2 and Fig. 3 b).
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUDING REMARKSAPPENDIX A: DERIVATION OF...APPENDIX B: VELOCITY FIELD...SUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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This work was supported by a La Ligue Contre le Cancer fellowship (to E.P.), a Human Frontier Science Program fellowship (to J.v.d.G.), a Curie Program Incitatif Coopératif grant, and a Human Frontiers Science Program grant (to C.S.).
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FOOTNOTES |
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E. Paluch's present address is Max Planck Institute of Molecular Cell Biology and Genetics, Dresden, Germany.
J. van der Gucht's present address is Laboratory of Physical Chemistry and Colloidal Science, Wageningen University, Wageningen, The Netherlands.
Submitted on April 27, 2006; accepted for publication July 12, 2006.
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONCONCLUDING REMARKSAPPENDIX A: DERIVATION OF...APPENDIX B: VELOCITY FIELD...SUPPLEMENTARY MATERIALACKNOWLEDGEMENTSREFERENCES |
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2 = 0.0019). Other continuous lines are to guide the eyes. (b) The average velocity of 1.5-µm radius beads as a function of the concentration of streptavidin in the presence of 2.5 µM biotinylated actin (30% of total actin) for gelsolin concentrations of 0.35 µM (solid diamonds), 0.54 µM (shaded triangles), 0.73 µM (squares), and 0.92 µM (light shaded down-triangles). (c) The effect of the cross-linkers filamin (squares), 
