Cells move by a dynamical reorganization of their cytoskeleton, orchestrated by a cascade of biochemical reactions directed to the membrane. Designed objects or bacteria can hijack this machinery to undergo actin-based propulsion inside cells or in a cell-like medium. These objects can explore the dynamical regimes of actin-based propulsion, and display different regimes of motion, in a continuous or periodic fashion. We show that bead movement can switch from one regime to the other, by changing the size of the beads or the surface concentration of the protein activating actin polymerization. We experimentally obtain the state diagram of the bead dynamics, in which the transitions between the different regimes can be understood by a theoretical approach based on an elastic force opposing a friction force. Moreover, the experimental characteristics of the movement, such as the velocity and the characteristic times of the periodic movement, are predicted by our theoretical analysis.
Eukaryotic cells move by a complex mechanism of extension and retraction controlled by the cytoskeleton dynamical assembly. During locomotion, cells extend protrusions at their leading edge, and form lamellipodia and filopodia, by rearranging their actin cytoskeleton. Active sites generate, at the plasma membrane, a polarized array of actin filaments that drives the membrane protrusions. Based on a similar molecular mechanism is the intracellular propulsion of lipid vesicles (1) and endosomes (2), and certain bacterial pathogens such as Listeria monocytogenes (3) and Shigella flexneri (4). In cell movement or in the movement of these vesicles or bacteria, the same or similar proteins are the key players for motility based on actin. Site-directed actin polymerization is induced at their surfaces, via the activation of the Arp2/3 complex by WASp (Wiskott-Aldrich Syndrome protein) family proteins (5) or by bacterial proteins (i.e., ActA for Listeria monocytogenes) (6). This spatially controlled polymerization results in the creation of a comet tail made of a dense actin gel. The force generated by the directionality of the actin filament growth is sufficient to propel these species.
The challenge in understanding the physical mechanism of force generation by actin assembly has given rise to various theoretical descriptions (7-12). For many years Listeria monocytogenes was used as a model system for studying the biochemistry of actin-based movement (3) and the effect of external influences on its movement (13). Listeria move at a velocity of a few µm/min, either smoothly or in a periodic fashion, as observed sometimes for wild-type bacteria (14) or genetically modified bacteria (15). However, the use of Listeria for testing theoretical models suffers from the drawback that geometrical parameters such as its size and shape are predetermined, and that the surface density of the Arp2/3 complex bacterial activator (ActA) is unknown. Biophysical studies on movement induced by actin polymerization were greatly facilitated by the development of in vitro systems that explore actin-driven motility of nonbiological cargos such as protein-coated beads (16) and lipid vesicles (17,18) placed in cell extracts. The discovery of the minimal set of proteins needed for reconstituting the movement of Listeria (19) opened the way for a controlled study of the physical parameters involved in actin-based motility (20). Such a system enables a detailed analysis of the physical parameters that govern actin-based movement. It was already experimentally demonstrated that at a fixed surface density of the actin polymerization activator, the type of movement is dramatically affected by changing the microsphere diameter, shifting it from a continuous to a jerky movement which resembles that of the mutated hopping Listeria (15).
In the work presented here, we investigate experimentally the bead diameter-protein surface density parameter space. Previously, only the influence of bead diameter was investigated in detail (20). We show that the transition from continuous to periodic regimes of motion can be induced simply by increasing the activating protein surface density as well as increasing the bead diameter. We further provide a scaling analysis of the transition from continuous to periodic regimes, and of the characteristics of this periodic regime. This analysis extends previous published works (10,21) that only provided numerical results relevant to the Listeria geometry. Here, we provide scaling guidelines for locating the transition from continuous to periodic behaviors and for characterizing the periodic regime. We finally compare these expectations to experimental results.
MATERIALS AND METHODS
Subdomains of the Wiskott-Aldrich Syndrome protein (WASP) and their purification
Two similar subdomains of WASP know as VCA and WA were bacterially expressed as GST fusion proteins as described in Bernheim-Groswasser et al. (20) for VCA and Fradelizi et al. (22) for WA.
Protein coating of polystyrene beads
Polystyrene beads of diameter ranging from 1 to 9.1 µm were purchased from Polyscience (Niles, IL). The error bars in bead diameter were given by the manufacturer (see Table 1). We varied the protein concentration on the surface by changing the protein concentration in the incubating solution. The beads were incubated in the protein solution at concentrations in the range of 0.01-1 mg/ml for 1 h at room temperature. We used an initial volume of beads that was proportional to the diameter, i.e., volumes of 1 µl and 10 µl for beads of 1 µm and 10 µm diameters, respectively, given that the volume fraction of the beads in the stock solution was constant, equal to 2.5%. The beads were finally resuspended in a constant total volume of 20 µl to ensure a constant total surface area independently of the bead diameter. The beads were stored at 4°C in a storage buffer (20 mM phosphate buffer pH = 7.4, 150 mM NaCl, 1.5 mM NaN^sub 3^, 10 mg/ml BSA, 5% glycerol) for up to 1 week.
Determination of the surface density of VCA proteins on the bead surface
The concentration of immobilized VCA was evaluated by SDS-PAGE analysis on washed VCA-coated beads heat-denatured in SDS. To determine the surface concentration C^sub s^ of actually active proteins, we used a pyrenylactin polymerization assay (20). Briefly, beads carrying different surface densities of VCA were placed in a solution containing 2.5 µM G-actin (10% pyrenyl-labeled) and 20 nM Arp2/3. A calibration was first obtained using known concentrations of soluble VCA. The concentration of active immobilized VCA on beads was determined by comparison with the polymerization rate of soluble VCA. Whereas the saturating conditions lead to a 1:1 ratio of active proteins, decreasing the surface concentrations resulted in decreasing the number of active proteins on the surface. As a result, the solution concentration of 0.2, 0.05, 0.035, and 0.02 mg/ml of proteins in the incubation solution corresponds respectively to concentrations C^sub s^ of (6 ± 1), (2.2 ± 0.3), (1.17 ± 0.02), (0.14 ± 0.02) × 10^sup -2^ mol/nm^sup 2^ active proteins on the bead surface. The error on the surface concentration measurement is of the order of 20% and includes bead loss in the bead-coating process due to repeated washing.
The motility medium contained 8.5 mM HEPES, pH = 7.7. 1.7 mM MgATP, 5.5 mM DTT, 0.12 mM Dabco, 0.1 M KCl, 1 mM MgCl^sub 2^, 6.5 µM F-actin, 0.1 µM Arp2/3 complex, 0.046 µM capping proteins, 2.5 µM ADF, 2.5 µM profilin. 0.54 µM α-actinin, 0.31% methyl-cellulose, 0.75% BSA, and 1.1 µM actin-rhodamin as optimized for Listeria movement (19). No vasodilator-stimulated phosphoprotein (VASP) was needed. A 0.3-µl volume of bead suspension was added to 20 µl of motility medium. The small volume of the beads ensured that the composition of the motility medium remained unchanged. The sample was placed immediately between a glass slide and an 18-mm-square glass coverslip sealed with vaseline/ lanolin/paraffin (at weight ratio of 1:1:1). To prevent squeezing of the objects, the distance between slide and coverslip was controlled using an inert polyemylene-glycol spacer (Goodfellow, Berwyn, PA) to obtain a ratio between sample height and bead diameter of 5:6. This ratio was found to ensure that the beads with their comets do not stick to the coverslip walls. In all experiments, symmetry breaking occurs spontaneously and 100% of the beads generate a comet.
Tracking and imaging of bead movement
The characterization of the bead movement, polymerization dynamics, and comet structure were performed by means of phase contrast and fluorescence microscopy techniques (Olympus 1X51 microscope, Melville, NY). Each data point was obtained by averaging experimental data from a minimum of 10 beads. The movement of the microspheres was tracked during 1.5 h by timelapse optical video-microscopy. Measurement of the beads velocity was performed as in Bernheim-Groswasser et al. (20). The thickness of the actin gel in the comet of saltatory moving beads was determined as the distance between the inflection points in the gray-level intensity curve obtained as a linescan of the middle of the comet.
Transition from continuous to saltatory movement
Beads of 1-9.1-µm diameter were grafted at different surface densities C^sub s^ of VCA or WA subdomain (C^sub s^, concentration of active proteins, from 0.14 to 6 × 10^sup -2^ molecules/nm^sup 2^, see Materials and Methods, above). Once placed in the motility medium, these beads started to assemble an actin gel, resulting in a homogeneous, spherical actin cloud around the bead. In a time ranging from 3 to 25 min, the spherical symmetry of the actin cloud was broken. This time (t^sub s^) is proportional to the bead size (20). From then on, an actin comet was developed and the beads were propelled forward. Two regimes of dynamics were clearly observed-i.e., continuous motion, and saltatory motion-depending on bead size or surface concentration of the activating proteins (Fig. 1). In the continuous regime, the comet developed behind the moving bead is dense and homogeneous except for its first layer (see inset A and C, Fig. 1), whereas the saltatory regime is characterized by a periodic motion of the beads, a non-constant velocity cycle, and a nonuniform comet density (see inset B and D, Fig. 1). In the saltatory regime, the dense and loose parts of the comet are respectively correlated with bead low- and high-speeds, as described in Bernheim-Groswasser et al. (20). From the experimental diagram in Fig. 1 we conclude that the bead movement depends on two controlled variables, the surface density C^sub s^ of active actin polymerization activators, and the bead diameter, D. The transition from continuous to saltatory movement can be induced by increasing either the bead diameter D or the protein surface concentration, C^sub s^. One can note that small beads (D ≤ 2.1 µm) always move continuously regardless of the surface concentration value C^sub s^, whereas larger beads (2.1
In practice, phase contrast imaging enables us to measure differences in gel density (gray-level) that appear in the cornet. If the bead is performing a continuous motion, the only gel density difference is between the first layer and the rest of the comet tail (see inset A, Fig. 1). However, if the bead is performing saltatory movement, there are variations in the comet gray-level (see Figs. 1 and 2). In both cases, when the polymerization begins (time t = 0), the bead is at rest and a gel layer is built until symmetry is broken (time t = t^sub s^) (20). The thickness of this first layer is found to be linearly dependent on the bead diameter (Fig. 2, triangles) and regardless of the motion type, whether it is continuous or saltatory. In the case of large beads (D > 2.1 µm), which perform discontinuous motion, the following layers are formed while the bead is already in motion: one gel layer is formed during each jump or velocity cycle (shaded arrows in Fig. 2). The thickness of these actin gel layers is found to be independent of the bead diameter, its value being e* = 1.5 ± 0.2 µm (squares and insets a-d in Fig. 2). One can note from Fig. 2 that the thickness of the first layer is always larger or equal to that of the following ones that are produced during the bead movement.
Experiment analysis (bead velocity, actin gel thickness)
The curvilinear distance traveled by the bead center as a function of time, X(t), was measured using video phase contrast optical microscopy. The velocity d/dtX(t) (time derivative of X(t)) was then calculated. The velocity of a 6.3-µm-diameter bead moving in a periodic motion is plotted in Fig. 3. The saltatory motion is characterized by a velocity cycle that starts at a maximal velocity V^sub max^, passes through a minimal velocity V^sub min^, and ends at V^sub max^. A careful look at a typical velocity cycle starting from V^sub max^ reveals three phases. The first two phases correspond to a steep decrease of the velocity to V^sub min^, followed by a slow increase of the velocity. These two phases are achieved in a total characteristic time τ' (Fig. 3), which is also the time needed for growing a gel of thickness e*. The third phase is a steep increase of the velocity to V^sub max^ and corresponds to the expulsion of the bead from the actin shell of thickness e*. This expulsion phase is achieved in a characteristic time τ (Fig. 3). For different bead sizes, we measured experimentally the characteristic time τ' required for growing a gel of thickness e* (see Table 1). We found that for beads of large diameter (D > 2.8 µm), the time τ' is always smaller than the time of symmetry breaking (t^sub s^), which is the time necessary to built the first layer. Note that this layer is always larger than e*. In the case of 2.8-µm-diameter beads, t^sub s^ equals τ' within experimental error and the thickness of the first layer equals the thickness of the other layers (see also Fig. 2). Additionally, we define as τ^sub rep^ the time that corresponds to the first phase (steep decrease of the velocity from V^sub max^ to V^sub min^). All times τ', τ, and τ^sub rep^ are defined in Fig. 3 and their experimental values are given in Table 1.
To analyze the transition from continuous to saltatory movement, we define the velocity amplitude (V^sub max^ - V^sub min^). The transition from steady to periodic movement is determined by plotting the normalized velocity amplitude (ΔV/V) = (V^sub max^ - V^sub min^)/(V^sub max^ + V^sub min^) as a function of the two control parameters of the system, D or C^sub s^. A typical first-order (discontinuous) transition appears in Fig. 4, where (ΔV/V) is plotted as function of the bead diameter D for a saturating activator surface density C^sub ss^. Small beads (D ≤ 2.1 µm) move smoothly, thus (ΔV/V) = 0. Increasing the bead diameter (D ≥ 2.8 µm) induces a transition to a motion type for which (ΔV/V) ≠ 0: for intermediate bead diameters (D = 2.8 µm) we observe the coexistence of two regimes of motion (solid dash-dot double arrow in Fig. 4)-a steady one, where (V^sub max^ - V^sub min^) = 0 and a saltatory one, with (V^sub max^ - V^sub min^) ≠ 0. In that case, the bead movement is intermittent, and shifts randomly from one velocity regime to the other. For larger bead diameters (D > 2.8 µm) the movement is always periodic, (ΔV/V) ≠ 0, and the velocity amplitude decreases with D (Fig. 4).
The transition from continuous to saltatory movement as function of D or C^sub s^ (Fig. 1) can be explained by the dimensionless elastic analysis proposed for the mechanism of Listeria monocytogenes propulsion (9,10). According to this analysis, the velocity (V) of the bead relative to the gel is given by the balance between the gel elastic propulsion force F^sub e^ and the gel/bead surface friction force F^sub f^, at any time. Notations used in the following sections are schemed on Fig. 5.
The elastic force
The friction force
Analysis of the bead movement
The bead velocity is determined by the balance between the elastic force f^sub e^ and the friction force f^sub f^(V) (in Fig. 6 and Fig. 7 these forces are normalized by the threshold force f^sub th^ defined below; also the velocities are normalized by the maximal velocity V^sub max^). Whenever the intercept of the two curves is far from the non-monotonous region of the f^sub f^(V) curve, the expected behavior is a simple steady motion (continuous regime), characterized by the velocity V^sub cont^, which is that of the intersection point (i.e., intercept f^sub f^(V) and f^sub e^(D^sub 1^) in Fig. 6 and intercept of f^sub f^(C^sub s2^) and f^sub e^(C^sub s2^) in Fig. 7). When the intercept is close to the non-monotonous region f^sub f^(V), the motion type cannot be directly deduced from our former analysis and depends on how fast the gel layer grows, while being expelled from the bead. In other words, when the elastic force f^sub e^ barely exceeds the threshold value f^sub th^ (Figs. 6 and 7), the friction drops abruptly and the gel layer is expelled from the bead in a characteristic time τ. If τ is long compared to the time τ' required for growing a gel layer around the bead, this gel layer is maintained, the system is at steady state and the bead motion is continuous at a constant velocity V^sub cont^ (Fig. 7). In the continuous regime, it is difficult to obtain an exact experimental estimate of the gel thickness around the bead. However, we observe that this thickness is not larger than the 1.5-µm thickness that corresponds to e* (see, for example, the continuous regime bead of Fig. 1). If τ is short compared to τ', the consequence will be the appearance of a saltatory motion where f^sub e^ [congruent with]f^sub th^ (see f^sub f^(V) and f^sub e^(D^sub 2^) in Fig. 6 and f^sub f^(C^sub s1^) and f^sub e^(C^sub s1^) in Fig. 7). For saltatory motion, a velocity cycle that starts at a maximal velocity V^sub max^ passes through a minimal velocity V^sub min^, and ends at V^sub max^. Rough estimates of the times τ and τ' gives τ [congruent with] D/V^sub max^ and τ' = e*/v^sub p^ (where v^sub p^ is the velocity of polymerization on the bead surface). The condition for obtaining a saltatory motion is thus V^sub max^ [much greater than] v^sub p^D/e*. Given that D > e* (see Fig. 2), the maximal velocity is always greater than the polymerization velocity. This is in agreement with our experiments since the maximal velocity is of the order of 1 µm/min (20), whereas the polymerization velocity (24) is lowered in the presence of stress (14).
According to the discussion of the preceding section, the increase in bead diameter D decreases the propulsion force per nucleator f^sub e^ (Eq. 3) without changing the friction term f^sub f^. Thus, varying D at constant nucleator surface concentration opens the possibility to investigate the state diagram of the system dynamics. The fact that small beads move at a relatively high constant velocity (~1 µm/min) is in agreement with the theoretical analysis of the force-velocity curve depicted in Fig. 6. One can see that the intersection of the elastic force f^sub f^(D^sub 1^) with the fast branch of the friction force f^sub f^(V) (c-d segment) results in a high and constant velocity. The increase of D reduces the propulsion force and drives the system toward the saltatory regime (f^sub f^(V) and f^sub e^(D^sub 2^) curves, Fig. 6) as observed experimentally (Figs. 1 and 4).
Conversely, varying the surface nucleator concentration at constant bead diameter also allows exploring the system state diagram. At a given velocity, the friction force per nucleator decreases when the nucleator density is increased: f^sub f^ is independent of C^sub s^, whereas the second term of Eq. 5 is inversely proportional to C^sub s^. As a consequence, the slope of the c-d part is a decreasing function of C^sub s^, as shown in Fig. 7.
The C^sub s^ dependence of the propulsion force is difficult to assess. A naive analysis would predict an increase of the elastic modulus with C^sub s^. However, under strong branching conditions controlled by the Arp2/3 complex, the elastic modulus Y should become essentially independent of C^sub s^. Thus the propulsion force per activator should increase upon decreasing C^sub s^. Thus, starting from a saltatory regime and decreasing C^sub s^ at constant D should lead the system to a continuous regime according to Fig. 7. Note that the stationary velocity V^sub cont^ in the continuous regime is smaller than the maximal velocity V^sub max^ in the saltatory regime in a suitable parameter range. This corresponds to our experimental observations, since V^sub max^/V^sub cont^ is 2.6 and 4.3 for 6.3-µm-diameter beads coated with 0.37 × C^sub ss^ and 0.02 × C^sub ss^ proteins, respectively, and 1.8 for 4.5-µm-diameter beads coated with 0.195 × C^sub ss^ proteins.
Analysis of the saltatory regime and numerical estimates
In practice we find that τ [much less than] τ^sub rep^
We investigated a simple experimental system for the study of the mechanism of actin-based propulsion to highlight the role of physical parameters at a mesoscopic scale. The state diagram of the system dynamics was studied as a function of the size of the beads and of the surface concentration of actin nucleators. Two distinct regimes of motion are observed-saltatory and continuous. The transition from one regime to the other is described with the use of a mesoscopic analysis that involves elastic and friction forces, and uses experimental estimates. A non-monotonous dependence of the friction as a function of the bead velocity explains the oscillatory behavior of the movement and well describes the experimental observations such as velocities and characteristic times. Its virtue is to cast the description of actin-based motility in a few parameters, which can be related to molecular mechanisms and measured experimentally.
We thank Marie-France Carlier for the gift of the motility medium, Sebastian Wiesner for his help in measuring the activity of the proteins immobilized on the surfaces, and Julie Plastino for her help on the experimental setup. We thank Franck Jülicher and Ken Sekimoto for many fruitful discussions.
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Anne Bernheim-Groswasser,* Jacques Prost,[dagger] and Cécile Sykes[dagger]
*Chemical Engineering Department, Ben-Gurion University, Beer-Sheva, Israel; and [dagger] Laboratoire Physicochimie "Curie", UMR 168, Institut Curie/Centre National de la Recherche Scientifique, Paris, France
Submitted November 5, 2004, and accepted for publication May 17, 2005.
Address reprint requests to Cécile Sykes, Tel.: 33-1-423-46790; E-mail: firstname.lastname@example.org.
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