Observation and Kinematic Description of Long Actin Tracks Induced by Spherical Beads

Abstract

We report an in vitro study comparing the growth of long actin tails induced by spherical beads coated with the verprolin central acidic domain of the polymerization enzyme N-WASP to that induced by Listeria monocytogenes in similar cellular extracts. The tracks behind the beads show characteristic differences in shape and curvature from those left by the bacteria, which have an elongated shape and a similar polymerization-inducing enzyme distributed only on the rear surface of the cell. The experimental tracks are simulated using a generalized kinematic model, which incorporates three modes of bead rotation with respect to the tail. The results show that the trajectories of spherical beads are mechanically deterministic rather than random, as suggested by stochastic models. Assessment of the bead rotation and its mechanistic basis offers insights into the biological function of actin-based motility.

Introduction

Eukaryotic cells move by coordinated reorganization of their dynamic cytoskeletal network. In this process, actin polymerization plays a crucial role in generating forces at the leading edge of the cell (1,2). Bacterial pathogens like Listeria monocytogenes and Rickettsia rickettsii exploit this actin-based motility inside infected host cells (3–6). These bacteria stimulate growth of dense filamentous actin comet tails, which propel the microbes for intracellular movement. The propulsion occurs as actin monomers are rapidly inserted near the rear surface of the bacterium while at any moment a nearly equal amount of the same protein is released from the tip of the tail. Therefore, the comet tail often remains constant in length as it continually pushes the bacterium forward.

A minimal set of essential proteins required for actin tail formation have been identified (7). This set of proteins includes actin-related protein 2/3 complex, an actin nucleation-promoting factor (NPF) such as ActA and WASP family proteins, a capping protein, and actin depolymerizing factor (ADF)/cofilin (7,8). Using a mixture of these proteins, or a cellular extract containing them, actin-based movement similar to that of Listeria has been reconstituted in vitro with a number of artificial cargos, such as polystyrene beads (8–10), phospholipid vesicles (11,12), and oil droplets (13) coated with NPFs.

Although there has been major progress in the biochemical characterization of actin-based motility, the biophysical mechanism of how a propulsive force is generated through actin polymerization remains a subject of ongoing study. In attempting to understand this mechanism, several models have been developed (14–19). Mogilner and Oster (15,18) proposed a tethered elastic Brownian ratchet model, postulating that the propulsive force is generated by the transient attachment and detachment of thermally fluctuating actin filaments to the surface of a moving cargo. Dickinson et al. hypothesize that the end-tracking motors hydrolyze ATP-actin to filamentous ADP-actin, thereby producing a force on the cargo (17). These models are proposed to explain the mechanism of actin-based motility at the molecular level, but they do not address the kinematics in regard to why actin-propelled cargos move in a variety of complex, sometimes periodic trajectories (3–6).

To understand actin-based motility at the microscopic or larger scale, several studies have shown how the curvature of trajectories depends on factors such as the cargo size and the density of actin filaments that push the cargos (20–25). Rutenberg and Grant (21) proposed a theoretical model of randomly oriented actin filaments propelling a bacterium. It is predicted that the resultant bacterial trajectories have curvature values that follow a Gaussian distribution centered at zero. Shaevitz and Fletcher (23) measured the curvature and torsion of long trajectories of RickA-coated beads in both two- and three-dimensional (2D and 3D, respectively) environments. Their trajectories showed highly varying curvature and torsion with a relatively short correlation time of ∼200 s, indicative of a growth process dominated by stochastic variation.

Recently, Shenoy et al. (25) have developed an analytical model with a set of deterministic equations that accounts for various seemingly unrelated 2D trajectories of Listeria in cytoplasmic extracts. With the geometrical constraint that under typical experimental conditions the sample is confined in a thin chamber, the model predicts 2D trajectories of Listeria with zero mean curvature in most cases, which agrees with observations by others (25). In unconstrained 3D motion of Listeria monocytogenes, the model predicts helical trajectories, also in agreement with a recent experimental finding with Listerial trajectories in calf-brain extract (26).

In this report, we show long trajectories induced by spherical beads with characteristic differences from those of bacteria in similar extracts. Our results show that 2D trajectories of beads often have nonzero mean curvatures. We explain this finding by incorporating an additional rotational term in the kinematic description, which accounts for the modified shapes and curvatures of 2D actin tracks induced by beads. In addition, we performed experiments with bead-induced actin tail growth in 3D, in which we observed trajectories of both right-handed and left-handed helices with nearly equal probability. Finally, we discuss implications of these new findings concerning the separate modes of bead rotation and the coupling between the bead surface and the actin tail, which raises mechanistic questions for future studies.

Kinematic description of actin tracks induced by spherical beads

In the kinematic description of 2D motion of Listeria (25), Shenoy et al. postulate a torque that arises from the rotation of the point of action of net propulsive force F around the long axis of the bacterium with an angular velocity ω. This net force generates a constant velocity v. At the point of action offset by a distance d from the axis, this force also produces a torque of constant magnitude, τ_⊥ = Fd, on the cell body (see Fig. 1 a), but the direction of the torque rotates in the plane perpendicular to the direction of bacterial motion. If the motion of the cell is confined in 2D (the xy plane), the model further assumes that the component of the torque projected to the plane of motion is counterbalanced by the planar constraint, whereas the orthogonal component normal to the plane of motion (along the z axis) gives rise to rotation of the cell body and subsequently causes curvature in the 2D trajectory. Based on this model, the angular velocity, $\stackrel{.}{\theta }$, of the bacterial body axis can be expressed as

$$\stackrel{.}{\theta }=d\theta /dt=\Omega \text{cos}\left(\omega \text{t}\right),$$ (1)

where Ω denotes the angular velocity of the bacterium for its rotation with respect to the z axis at t = 0, conveniently defined as when the point of action of the net force lies in the xy plane where the axis of the bacterium lies. The expression above tracks the rotation rate of the tangent along the trajectory as the bacterium is propelled forward with a constant speed v. Since the curvature of the trajectory is $\kappa =\mathrm{d}\theta /\mathrm{d}\mathrm{s}=\dot{\theta }/\mathrm{v}=(\Omega /\mathrm{v})\text{cos}\left(\omega \text{t}\right)$, the mean curvature κ_m over a long trajectory becomes zero (see Note S1 in the Supporting Material).

Figure 1.

Kinematic model of 2D motion of Listeria monocytogenes in comparison with that of a spherical bead. (a) The net propulsive force, F, acting on a Listeria by its actin comet tail remains parallel to the long axis of the bacterium at an offset d, which rotates with an angular velocity ω around the axis. This results in a constant torque, τ_⊥ = d × F, that is perpendicular to the plane of motion, but the torque direction also rotates around the cell axis. Here, θ denotes the orientation of the bacterium, and v is its velocity of motion. The dotted line indicates the trajectory of the bacterium. (Adapted with modifications from Fig. 2 of Shenoy et al. (25).) (b) A spherical bead can rotate not only around the axis that passes through the center of the tail, but also around the z axis (pointing toward the reader). The force F′ depicts the portion of the force generated by the actin tail that causes an additional rotation about the z axis with angular velocity β. This rotation is uncoupled from the bead rotation around the direction of the growth trajectory (ω), which is the property commonly illustrated for F in Fig. 1, a and b. A nonzero β allows for nonzero mean curvature in the resultant trajectories of beads. (c) A simple illustration to relate bead rotation about the z axis with the curvature of the actin track (green). Illustrations of Fig. 1, b and c, assume a non-slip condition between the bead and the actin tail. Therefore, when the trajectory completes one circle, the bead rotates with respect to the z axis by exactly 360°, as illustrated by the dotted white arrows.

Our model describing the bead-induced actin track evolves from what is summarized above based on the previous work (25). First, we modify the language from talking about rotating force and torque to, more explicitly, the driven rotation of the bead. The physical basis of this more explicit language is that the actin tail is known to be mechanically attached to the cargo, be it a bacterium (14), an enzyme-coated bead (27), or a lipid vesicle (12). Second, we add an additional term of rotation for the case with beads, initially put forth by Tambe (28). It is important to note here that we provide in this work a physical interpretation of the additional rotation term that differs from that envisioned in Dr. Tambe's thesis. This additional rotation term plays a key role in explaining the generalized kinematic model.

To interpret 2D trajectories induced by enzyme-coated beads, we start with the assumption that the growing actin network causes rotations of the bead with respect to axes along its trajectory and normal to the plane of motion (the z axis, conveniently chosen as going through the bead center), noted as ω and $\stackrel{.}{\theta }$, respectively. Based on the continuous attachment of the actin tail to the bead surface (27,29–31), we hypothesize that there is no rotational slippage of the bead at its interface with the growing actin tail. (The model can be generalized even if slippage is allowed, although the existing data referred to here have proven this scenario to be unrealistic. See Note S2 in the Supporting Material). With the attachment of the growing actin tail to the bead surface, the shape and curvature variation of the trajectory are totally prescribed by the bead rotation along the z axis, $\stackrel{.}{\theta }$. Specifically, the bead rotation along the z axis consists of two terms: one coupled to the rotation about the direction of growth (ω), which is the same as the cell axis for the case of Listeria, illustrated in Fig. 1 a as being pushed by F, and an additional rotation of the bead about the z axis, illustrated in Fig. 1 b as being pushed by F′. When these two terms are incorporated, the new equation of bead motion becomes

$$\stackrel{.}{\theta }=\Omega \text{cos}\left(\omega \text{t}\right)+\beta ,$$ (2)

where ω and Ω have the same meaning for bead rotation as introduced above for Listeria, but β denotes an additional rate of rotation of the bead about the z axis. Unlike Ω, β is decoupled from the rotation about the direction of growth, ω. The mechanical consequence of β alone is shown in Fig. 1 c, where a clockwise rotation of the bead gives rise to a circular actin track, turning clockwise at the same rate as the bead rotation, β. In the case of Listeria, the first term of Eq. 2 appears sufficient to explain the trajectories they induce, suggesting that β = 0 (25). In the case of spherical beads, however, both terms in Eq. 2 contribute to the rotation of the bead about the z axis. In Note S3 in the Supporting Material, we offer two plausible physical pictures to help differentiate β from Ω.

Based on Eq. 2, the curvature of a bead trajectory becomes $\kappa =\stackrel{.}{\theta }/\mathrm{v}=(\Omega /\mathrm{v})\text{cos}\left(\omega \text{t}\right)+\beta /\mathrm{v}$, so that the long-term mean curvature becomes a constant, β/v. This analysis makes it clear that β results in nonzero mean curvature of the 2D bead trajectory.

By integrating Eq. 2, the orientation of the bead becomes

$$\theta =(\Omega /\omega )\text{sin}\omega \text{t}+(\beta /\omega )\omega \text{t}+{\theta }_{\text{0}},$$ (3)

where θ₀ is a constant of integration, representing the initial angle of orientation when t = 0. The physical picture of this model is that the bead is pushed by the growing tail with a constant speed, and the growth direction varies as a function of time (Eq. 3).

This actin track led by the bead is fully described by the translational motion of the bead center. Specifically, the velocity vector of the bead center, $\stackrel{.}{\mathbf{r}}=(\stackrel{.}{\mathrm{x}},\stackrel{.}{\mathrm{y}})=\text{v(cos}\theta ,\text{sin}\theta )$ can be written as

$$\stackrel{.}{x}=\text{v}\text{cos}[(\Omega /\omega )\text{sin}\omega \text{t}+\left(\beta /\omega \right)\omega \text{t}+{\theta }_{\text{0}}],$$

$$\stackrel{.}{y}=\text{v}\text{sin}\left[\left(\Omega /\omega \right)\text{sin}\omega \text{t}+\left(\beta /\omega \right)\omega \text{t}+{\theta }_{\text{0}}\right].$$ (4)

The bead trajectory, (x(t), y(t)), can then be obtained by integrating $\left(\stackrel{.}{x}\right(\mathrm{t}),\stackrel{.}{y}(\mathrm{t}\left)\right)$ over time, giving rise to various shapes. In particular, the additional term with the parameter β/ω produces a richer variety of shapes than Listeria, as shown earlier (25).

Materials and Methods

Proteins and extracts

Actin was extracted and purified from rabbit skeletal muscle as previously described (32). The purified actin monomers were equilibrated with G-buffer (2 mM Tris-HCl, pH 8.0, 0.2 mM ATP, 0.5 mM DTT, and 0.2 mM CaCl₂). The protein was rapidly frozen in liquid nitrogen and stored at −80°C. Alexa 488-labeled G-actin was prepared according to the procedure in Fisher and Kuo (33), using Alexa 488-succinimidyl ester (Invitrogen, Carlsbad, CA). ADF/cofilin and the verprolin central acidic domain (VCA) of N-WASP were purchased from Cytoskeleton (Denver, CO). Platelet extracts were prepared from freshly outdated plasma rich in unstimulated human platelets, as described by Laurent and Carlier (34). The platelet lysates were obtained by sonication in a buffer (10 mM Tris-HCl, pH 7.5, 20 mM EGTA, and 2 mM MgCl₂) supplemented with 1 mM phenylmethanesulfonyl fluoride and protease inhibitors (10 μg/ml each of leupeptin, pepstatin, and chymostatin). After sonication, the lysates were centrifuged at 100,000 g for 45 min at 4°C to sediment membrane fragments. The supernatant was supplemented with 1 mM ATP, 1 mM DTT, and 150 mM sucrose, rapidly frozen in liquid nitrogen, and stored at −80°C. Before experiments, the frozen aliquots were thawed to room temperature by a water bath.

Beads and motility assay preparation

Carboxylated polystyrene beads (D = 1 μm and 2 μm; Polysciences, Warrington, PA) were coated with VCA as described in van der Gucht et al. (35). First, the beads were incubated with 10 μM VCA solution for 1 h. Second, 10 mg/ml bovine serum albumin (BSA) was added and the sample was incubated on ice for an additional 15 min to block free adsorption sites on the beads. VCA beads were then centrifuged and washed twice with a washing buffer (10 mM Hepes, pH 7.8, 100 mM KCl, 1 mM MgCl₂, 1 mM ATP, and 0.1 mM CaCl₂). Finally, the VCA beads were stored in a storage buffer (10 mM Hepes, pH 7.8, 100 mM KCl, 1 mM MgCl₂, 1 mM ATP, 0.1 mM CaCl₂, and 1 mg/ml BSA). The freshly coated beads were used for up to 3 days.

For the motility experiment, VCA-coated beads (2 μl) were mixed with 10 μl platelet extract containing extra G-actin monomers (∼6.25–12.5 μM in final concentration) and 1 μl of 10× ATP regeneration mix (20 mM ATP, 100 mM creatine phosphate disodium salt, and 35 units/ml creatine kinase). For imaging by fluorescence microscopy, Alexa-488 labeled G-actin was used instead of unlabeled actin monomers. To produce long trajectories, additional 0.2% BSA and 5 μM cofilin were added into the standard motility assay. For a 2D motility chamber, the sample was gently mixed in an Eppendorf tube and 2 μl aliquot was sandwiched between a BSA-treated coverslip (22 mm × 22 mm) and a glass slide, sealed with vacuum grease for microscopy observation. To prepare a 3D chamber, we used double-sided tapes between a coverslip and a glass slide to contain a 10-μl sample. The measured average thicknesses of the 2D and 3D chambers were ∼5 μm and 120 μm, respectively. The in-plane sample size was always larger than a spot 1 cm in diameter. No constraint was imposed from the sides as the microscopy observation was made away from edges.

Microscopy imaging

Both phase contrast and fluorescence images of actin tails were taken using a Nikon E800 microscope equipped with a Cool Snap HQ camera (Photometrics, Tucson, AZ) with a 60× Nikon oil immersion lens. Confocal microscopy was performed with a Leica TCS SP2 confocal laser scanning microscope (Leica Microsystems, Mannheim, Germany) coupled to a Leica DM IRE2 inverted microscope and equipped with argon and helium neon lasers. Alexa-488-labeled actin was imaged by an Ar-Ne laser with the excitation wavelength of 488 nm. To visualize the 3D structure of the actin tails, a series of confocal images with step size Δz = 0.12 μm was obtained to generate z stacks. With these z-stacks, 3D animations were created. The movie documents were generated from the image stacks stored at 10 frames/s as AVI files. All color-coded images and 3D animations were rendered using the Leica confocal software.

Curvature analysis

Each 2D image was traced by hand to reduce the trajectory to a simple line. These lines were digitized using MATLAB's Image Processing Toolbox to produce a faithful xy-trajectory dataset. To remove pixilation noise, the trajectories were smoothed by repeatedly averaging each coordinate with its nearest neighbors. Long trajectories were then resampled every 1–2 μm to focus only on large-scale curvature, thus further minimizing the effect of local kinks in the measured trajectories, often due to noises in the images. Curvature was calculated by taking the inverse radius of the best-fit circle to the curve for each group of three or more data points. For greater accuracy, more points were used in regions of higher curvature. The 3D trajectories were analyzed as detailed in Note S4 in the Supporting Material, yielding both curvature and torsion.

For each 3D trajectory, several 2D confocal slices were thresholded and digitized using MATLAB's Image Processing Toolbox to reconstruct a 3D image volume. Adjacent clusters of points were enclosed in spheres of a radius larger than the cross-sectional diameter of the tubular 3D image. Coordinates of points within a sampling sphere were averaged to acquire a centroid. A number of centroids were connected to yield a trajectory that passed through the image volume. This method can be applied to 2D trajectories as well, using circles rather than spheres to obtain centroids. In the 2D case, however, we found that hand-tracing worked as reliably and was more convenient. In fact, the centroid tracing method was applied for 3D trajectories only because the hand-tracing method was no longer possible.

The 3D trajectories were smoothed using moving averages and were resampled in a manner similar to that in the 2D case. The curvature and torsion at each point were calculated by the Frenet-Serret formulas, which describe the curvature and torsion of a 3D curve by the magnitude of the instantaneous rates of change of its tangent (T), normal (N), and binormal (B) vectors. Specifically, the vectors are defined as shown below (36):

$$\mathbf{T}=\frac{d\mathbf{r}}{d\text{s}},\mathbf{N}=\frac{\text{d}\mathbf{T}/\text{ds}}{|\text{d}\mathbf{T}/\text{ds}|},\mathbf{B}=\mathbf{T}\times \mathbf{N,}$$ (5)

where r is the position vector of the trajectory and s is its cumulative contour length. These form a rotating, 3D orthonormal basis at each point along the curve. Then, curvature (κ) and torsion (τ) are defined as

$$\frac{\text{d}\mathbf{T}}{\text{ds}}=\kappa \mathbf{N},\frac{\text{d}\mathbf{B}}{\text{ds}}=-\tau \mathbf{N}\text{.}$$ (6)

In the actual trajectory analysis, all derivatives were approximated by first differences.

Parameters of trajectories

Two procedures of fitting can be applied to obtain parameters for simulated trajectories closely resembling those observed. The first follows what has been described previously (25). In this method, subroutines from MATLAB's image-processing tool box are used to generate fit trajectories from Eq. 4 that closely match those observed, yielding v, ω, Ω/ω, and β/ω. The second method involves a direct fit to an expression derived from Eq. 2 after the curvature is obtained as a function of contour length L, κ = (Ω/v)cos(2πL/λ) + β/v, where λ is the wavelength on the κ-L plot. By fitting data to this formula, one obtains v from the expression v = L/t, where t is the growth time corresponding to L, Ω from the amplitude, β from the constant offset, and ω from ω = 2πv/λ. We chose the first method to avoid noise introduced in our curvature calculation. The drawback of this choice is that the average curvature obtained from the simulated trajectory fit need not be exactly the average value obtained from the κ-L plot, although our results show that the two values agree with each other rather well.

Results and Discussion

Biochemical conditions for generating long trajectories of beads

Trails of a dense actin network induced by VCA-coated beads were conveniently observed as 2D trajectories as the samples were typically confined in 5 μm thick motility chambers. We produced long actin tails in the platelet extract, supplemented with actin and ATP regeneration mix, using 1 μm beads uniformly coated with VCA. The choice of platelet extract was made according to the availability of the starting material to us, and the activities of the extract prepared for bead-induced motility proved to work as well as those in Xenopus eggs (25) or rat brain (26) extracts used by others. Initially, symmetric actin clouds formed around VCA-coated beads. After breaking off from symmetric shells, VCA-coated beads induced growth of actin tails, which grew over several hours to form a variety of curved trajectories. The typical time to break symmetry was ∼10–15 min. Sample phase contrast images of actin tail growth are shown in Fig. S1 a in the Supporting Material (see also Movie S1). The average speed of the bead was 86 nm/min. This speed was also the average tail-growth rate induced by this bead, since in our study all the long tails remained stationary and did not depolymerize over the course of many hours. This might be due to either a high concentration of cross-linking proteins preexisting in the platelet extract, or a lack of ADF. When the bead ran into an obstacle (in this case the existing actin tail), it slowed down to 27 nm/min, as shown in Fig. S1 b. However, after passing through the obstacle, the bead recovered its faster initial speed.

To produce long trajectories, we sought to optimize the biochemical conditions, which led to a faster tail-growth rate that persisted over a long period of time. First, we observed that the average tail-growth rate increased about twofold as concentrations of rabbit skeletal muscle G-actin added to platelet extract increased from 0.25 to 0.5 mg/ml. Curiously, this result differs from a previous finding that the addition of rabbit skeletal muscle actin decreased the speed of 0.5-μm ActA-coated beads in Xenopus egg extract (20). This difference might be attributable to the two different types of extracts used. Second, we found that addition of either BSA or ADF/cofilin also accelerated actin tail growth. Adding both proteins produced the fastest growth (Fig. S2), with an average speed of 0.60 ± 0.05 μm/min. BSA blocks the nonspecific adsorption of beads and actin tails to the glass surface, which might hinder the growing separation between the two when actin monomers insert at the base of the tail, thereby retarding the tail growth. The positive effect of added ADF/cofilin on growing long actin tails is not as intuitive. It has been reported, however, that ADF/cofilin acts synergistically with actin-related protein 2/3 complex in vivo to promote cell protrusion (37), which may be relevant to our finding.

Experimental trajectories can be simulated with appropriate rotation rates

Long 2D trajectories with a variety of shapes were generated under the optimal condition, as specified above, and imaged by both phase contrast (Fig. 2, a–d) and confocal microscopy (Fig. 2, e–h). We analyzed these trajectories by applying the kinematic model. For this analysis, we calculated the curvatures of the observed trajectories in Fig. 2, and then obtained the parameters for simulating these trajectories using the methods described above. Representative examples of the curvature analysis and simulation from images in Fig. 2 e, c, and a, are shown in Figs. 3–5, respectively. In all three cases, the simulated trajectories resemble the periodic features of their experimental counterparts and prescribe mean curvature values that agree with experimental values.

Figure 2.

Long 2D trajectories of 1-μm VCA-coated spherical beads in platelet extract with various shapes. (a–d) Phase contrast images. (e–h) Color-coded 3D rendition of confocal images obtained by a series of confocal images with a step size of Δz = 0.12 μm to generate a z-projection of stacks based on volume (scale bar, 10 μm). The color bar corresponding to the confocal images indicates the limited trajectory depth in the z-direction. The trajectories shown here generally have nonzero mean curvatures, showing a crucial difference between beads and Listeria trajectories. The sample condition is a VCA-coated bead in platelet extract supplemented by 9.5 μM G-actin (0.4 mg/ml), 0.2% BSA, 5 μM Cofilin, and ATP regeneration mix.

Figure 3.

(a) An enlarged segment of the phase-contrast image of a trajectory (clipped from Fig. 2e) showing a counterclockwise, closed path (black arrowheads). The blue line is a trace of the trajectory, with maximal and minimal points (A–C) of the local curvature indicated. (b) The best-fit theoretical trajectory corresponding to the experimental trajectory. A′, B′, and C′ are counterparts of A, B, and C, respectively, in a. (c) Analyzed curvature of the experimental trajectory, plotted as a function of contour length. Mean curvature is ∼0.04 μm⁻¹ (dotted line). (d) Curvature plot of the theoretical trajectory in Fig. 3b.

Figure 4.

(a) Phase-contrast image of a trajectory (see Fig. 2c) showing a loopy feature (direction of motion indicated by white arrowheads). The blue line is a trace of the trajectory, with points A–C indicating where the curvature is maximal or minimal. (b) A theoretical trajectory corresponding to the experimental trajectory in Fig. 4a. A′, B′, and C′ are counterparts of A, B, and C, respectively. (c) Analyzed curvature of the experimental trajectory, plotted as a function of contour length. Mean curvature is -0.3 μm⁻¹, denoted by a dotted line. (d) Curvature plot of the theoretical trajectory in Fig. 4b.

Figure 5.

(a) Phase-contrast image of a trajectory (Fig. 2a) characterized by a long path with two sets of loops (direction of movement indicated by arrowheads). The blue line is a trace of the trajectory, with points A–C showing where the curvature is maximal or minimal. (b) A theoretical trajectory corresponding to the experimental trajectory, where A′, B′, and C′ are counterparts of A, B, and C, respectively, in a. (c) Analyzed curvature of the experimental trajectory, plotted as a function of contour length. Mean curvature is −0.1 μm⁻¹ (dotted line). (d) Curvature plot of the theoretical trajectory in Fig. 5b.

Selected trajectories show distinct physical features corresponding to different values of the rotation rates ω, Ω, and β, all within the same kinematic model for spherical beads. The bead trajectory shown in Fig. 3 illustrates a counterclockwise path with undulating curvatures. The measured mean curvature of the trajectory is 0.04 μm⁻¹. The points A, B, and C and A′, B′, and C′ are marked as counterparts to guide the eye in the direction of movement, as well as to indicate where local curvatures are at a maximum or minimum. The simulated curve adopts parameter ratios of β/ω = 0.36 and Ω/ω = 0.48. Another bead trajectory is demonstrated in Fig. 4, which revolves in clockwise loops. The curvature of this trajectory is always negative, with an average of −0.3 μm⁻¹. We explain this nonalternating sign of κ by noting the condition of β > Ω (see Table 1). It is informative to contrast this trajectory with that shown in Fig. 3, where β < Ω and the curvature alternates sign locally (for instance, near point B in Fig. 3 a) while maintaining an overall counterclockwise motion. In Fig. 5, we present a trajectory with two sets of loops, connected by a more or less straight line. The curvature with a mean value of −0.1 μm⁻¹ only switches sign briefly over the entire trajectory. The simulated trajectory adopts comparable values of β and Ω and a larger value of Ω/ω (2.60) compared to those in Fig. 3 (0.48) and Fig. 4 (0.56). As a result, between the two double loops, there is a large segment of the trajectory over which the curvature is nearly zero, i.e., the bead travels along a more extended and delocalized path. A similar feature is shown in Fig. 2 b with two relatively straight paths, each connecting two sets of loops. The case shown in Fig. S4 is distinct in the sense that the entire trajectory encompasses only a fraction of a period corresponding to ω. Due to the large ratio of Ω/ω = 20, the curvature varies over a wider range. Along the trajectory, an initially gently curved path becomes progressively more curved, ending with tightening loops.

Table 1.

A comparison of shapes and fitting parameters for simulated trajectories

Trajectory source	Shape features	v (μm/min)	ω (rad/s)	Ω (rad/s)	β (rad/s)	Ω/ω	β/ω
Fig. 3	Large undulating loop; heartlike shape	0.24	0.021	0.01	0.0076	0.48	0.36
Fig. 4	Looped loops; no change in sign of curvature	0.18	0.05	0.028	-0.053	0.56	-1.07
Fig. 5	Loops-straight line-loops; path delocalized	0.24	0.01	0.026	0.029	2.60	2.29
Fig. S3	Spiral; tightening loop; curvature rising over time	0.6	0.02	0.401	0.34	20.05	17.00
Fig. S4	Folded loops; alternating curvature	0.068	0.01	0.028	0.0002	2.80	0.02

Comparison between beads and Listeria tails

The long 2D trajectories of beads exhibit notable differences from those of Listeria. All the bead trajectories analyzed above yield nonzero mean curvatures, different from previously observed Listeria trajectories under similar conditions (3–6,25). In particular, the bead curvatures do not necessarily alternate in sign from one loop to the next (as shown in Fig. 2, a, c, e, and g), which have not been observed in Listeria tails. As described in Eq. 2, the propulsive forces acting on the bead allow for two separate rates of rotation about the z axis, Ω and β, with the latter uncoupled from the rotation with respect to the direction of bead motion (ω). The nonzero mean curvature for bead trajectories arises from nonzero β. For the case of the Listeria trajectories, β = 0, as shown by the previous work (25).

Another notable feature of the bead-induced trajectories is the less smooth, more ragged appearance of their contours compared to Listeria trajectories. The noise and variation in curvature are clearly notable on all the κ-L plots (Figs. 3 c–5 c, Fig. S3 c, and Fig. S4 c). In contrast, the simulations are made using fixed values of ω, Ω, and β. Our treatment in capturing the key features of the observed trajectories ignores the complication that considerable variation of all three fit parameters would be required to reproduce the experimental trajectories. In some cases, since the shape changed abruptly at certain locations, a segment of a long trajectory had to be ignored to avoid dealing with a drastic change of any of these fit parameters. For instance, Fig. 2 e was clipped for analysis, as shown in Fig. 4, which ignored the looped part of the trajectory. The clear shape change occurred as the bead crossed its own path, which caused sudden changes of the rotation rates. Our simplified treatment falsely assumes these parameters to remain constant along the trajectory and thus could not simulate the trajectory in its entirety. Note a similar property in Fig. 3 b, where the curvature changes whenever the bead passes near or through the existing actin tail. These observations show that the actin network comprising the comet tail can be altered quite readily when the cargo encounters an obstacle, which acts as a perturbation to the network. Even discounting these abrupt changes, the noisy appearance in all the κ-L plots clearly points to the less deterministic nature of bead trajectories in comparison with the smoother and better defined Listeria trajectories (25).

Finally, we note that the bead-induced actin tracks also show certain similarities to Listeria trajectories, especially when β << ω. Fig. 2 d shows an example of a closed orbit with alternating curvature (β/ω = 0.02), with a mean curvature of −0.07 μm⁻¹. Fig. S4 shows the curvature of this trajectory side by side with a best-fit trajectory simulated from the model. The main features of this trajectory are captured in the simulated trajectories for Listeria, as shown in Fig. 3 of Shenoy et al. (25).

In summary, the shape of a bead-induced trajectory sensitively depends on the rotational rates ω, Ω, and β, as predicted by the kinematic model. As described in the Materials and Methods section, we obtain v, ω, Ω/ω, and β/ω from fitting the simulated trajectories to those observed experimentally. We then calculate Ω and β and compare these six parameters for all the trajectories, which are listed in Table 1.

Helical trajectories formed in 3D chambers

When constraints for 2D planar motion are removed, 3D helical trajectories of spherical beads are produced, as predicted by the kinematic model. Fig. 6 shows confocal microscopy images of four 3D helical actin tails of VCA-coated beads formed within a 120-μm-thick motility chamber. We observed eight right-handed and seven left-handed helices out of n = 15 trajectories, concluding that neither handedness is strongly favored. These results agree with a recent observation (23), but are different from results of another study of helical tails of Listeria, which shows only right-handed helices (26). We then measured curvature (κ) and torsion (τ), calculated using Frenet-Serret formulas after acquiring digitized 3D trajectories (see Fig. S5 and Fig. S6) (36). Both κ and τ vary somewhat over the long trajectories. For comparison, we overlay the actual trajectories with ideal helices of κ and τ adopted from the average values of the observed trajectories. The helical trajectory is deterministic, in contrast to the trajectories observed in the previous study by Shaevitz and Fletcher (23), showing a short curvature correlation length of ∼16 μm. In our observation, the correlation decay length becomes comparable to or even larger than the contour length of the trajectory. It can no longer be determined using the same criterion, which is statistically meaningful only if the counter length is orders of magnitude longer than the correlation length. Note that the study by Shaevitz and Fletcher was performed in Xenopus laevis egg extract, whereas ours used platelet extract. A more notable difference is that their study used RickA to coat the beads, whereas ours used VCA, which might cause a tighter mechanical link between the actin tail and the bead. A recent model by Dickinson suggests that the actin growth tightly connected to the cargo might yield a more deterministic trajectory, whereas the one loosely connected might allow more variation in κ and τ (38). Earlier studies have shown that RickA actin tails are less dense and consist of long, unbranched actin filaments rather than branched, cross-linked, dense network (39,40). The difference in the actin tracks formed by beads coated with these two NPFs may be attributable to differences in the molecular interaction and microscopic structure of the actin tails they form.

Figure 6.

A collection of 3D helical actin tails formed on VCA-coated beads (D=2 μm) in a 120 μμm thick motility chamber, imaged by confocal microscopy (Scale bar = 10 μm). Both right handed (Fig. 6, a and c) and left handed (Fig. 6, b and d) helices were observed with nearly equal probabilities. The color bar denotes the trajectory depth in z-direction with the actual z dimensions of 0-25 μm (Fig. 6, a and b) and 0-15 μm (Fig. 6, c and d).

The curvature and torsion of the Listeria trajectories in 3D have been derived as κ = Ω(t)/v(t) and τ = ω(t)/v(t) (see Supporting Information in Shenoy et al. (25)). Therefore, constant Ω and ω will give rise to constant κ and τ, as in an ideal helix. Despite the noticeable helical shapes in all 3D bead trajectories in our study, we also note considerable variation in κ and τ along each trajectory (see Fig. S5, c and d, and Fig. S6, c and d). Since we have not included in this work a kinematic analysis for 3D bead trajectories, it is hard to compare the extent of these variations with those reflected in the 2D κ-L plots. We have less understanding of the possible contribution of β, if any, to the 3D bead trajectories. Bearing these in mind, the noises noted in the κ and τ plots in Fig. S5 and Fig. S6 correspond to perhaps comparable variance in the 2D analysis. We admit here that in our kinematic model, Ω and β have not been properly defined in the 3D case for bead trajectories, so further analysis is needed for a reliable comparison.

Mechanistic implications of the kinematic model

Although the molecular mechanism that underlies our kinematic model is not fully established, the recent modeling works by Dickinson (38) and Lin et al. (41) provide a plausible account of its main assumptions, namely persistent longitudinal rotation and helical trajectories. Dickinson suggests that actin polymerization at the rear surface of an elongated cell body may be diffusion-limited so that growth near the outer edge of the actin tail is more favorable than that near the axis of the tail due to depletion of actin monomers inside. The resultant stress imbalance may lead to longitudinal rotation so that at the steady state, the growth rate on the outside is larger than that inside. Furthermore, if the NPFs are distributed asymmetrically around the bacterial axis, the point of action of the net force will be offset from the axis and it would rotate around the axis along with the cell body. This would lead to a helical trajectory in 3D, or a variety of 2D trajectories predicted by the kinematic model for Listeria trajectories (25). Note that the main assumptions of the Dickinson model remain valid for bead-induced actin trajectories, provided that the growth is polarized to one side of the bead due to possibly nonuniform coating, shell cracking caused by hoop stress (35), or any other mechanism. Therefore, the physical basis for both ω and Ω is reasonably sound.

The additional rotation rate, β, is key to the analysis of bead trajectories, but its physical mechanism is less defined. We attempt to understand β by comparing the bead trajectories with those of Listeria, in which case β is assumed to be zero. The most obvious difference between beads and Listeria is geometry. The spherical shape of the bead allows easy rotation with respect to any axis. In contrast, for the elongated Listeria cell body, the rotational moment along its long axis is much less than along either orthogonal axis. In addition, the localized distribution of ActA near the rear pole of Listeria imposes a geometrical constraint (42,43). Therefore, Listeria trajectories tend to be smoother than bead trajectories, allowing for only small variation of Ω. Second, the necessity of adding a β term possibly also depends on how tightly the tail structure is attached to the cargo. Specifically, the Dickinson model predicts that stronger attachment is expected to lead to more persistent longitudinal rotation and thereby more deterministic trajectories (17,38). A Listeria bacterium is encapsulated by the actin tail via localized distribution of NPF, so the tail attachment to the cell surface might be tighter than between an NPF-coated bead and its actin tail. Since the actin tail behind the bead tends to be more loosely attached, the extra rotation term (β) may occur. If the attachment becomes too weak, the trajectory might become too random to apply the kinematic model. Therefore, to what extent the model is applicable depends a great deal on experimental conditions, specifically, the strength of the attachment of the actin tail to the moving cargo. For instance, the experiments using dead Listeria in concentrated Xenopus cytoplasmic extract might be a good case of strong attachment (25). In contrast, a different study using polystyrene beads coated with purified RickA, also in Xenopus extract, might be a case of relatively weak attachment, as argued earlier, with the observed trajectories much more random (23). The VCA-coated polystyrene beads in our study might result in stronger attachment than with RickA, thus we found actin trajectories with a reasonably good fit to the generalized kinematic model, although the trajectories are not nearly as smooth as that shown for dead Listeria in Xenopus extract (25). In short, the findings of our study suggest a practical rationale for those seemingly contradictory findings (23,25).

Existing evidence for the kinematic model is somewhat mixed. Rotation of Listeria around the long axis (ω) has been observed by attaching a small fluorescent bead to the bacterial surface (37). Such experiments have not been done together with curvature analysis for cases of complex, curved trajectories to directly link the local curvature with cell body rotation. In the case of beads, a recently published study by the same group suggests that the local curvature of the actin track is not tightly coupled with the bead rotation, although the actual trajectories are not shown (20). Recognizing that the rotation of a bead is less restricted due to its spherical geometry, as well as weaker attachment with its actin tail, we analyze the curvature and shape of the trajectories, which strongly suggest the presence of these separate terms of rotation during its forward motion. Further experiments, hopefully with improved microscopic resolution, are needed to confirm the bead rotation as suggested by the model.

Conclusions

This article reports observation of long (up to 300 μm) and periodic trajectories induced by VCA-coated spherical beads in platelet extracts. We demonstrate that long actin tracks induced by the VCA-coated beads can be simulated by a kinematic model, incorporating a constant net force and persistent rotation of the bead to define the observed trajectories. The larger variety of shapes of actin tracks induced by spherical beads follows naturally from an extension of the 2D Listeria model, by adding an additional term of rotation in the kinematic model developed previously (25). Like the Listeria trajectories, the bead trajectories show mechanically deterministic shapes, with periodic rather than randomly varying curvature and torsion, as suggested by the stochastic growth models (22,23). Despite the complex dynamics of actin network growth, the trajectories remain steady over long periods of time. These results suggest that the actin network rectifies growth imbalance likely related to its loading history (44) and thereby maintains stable movement. We also envision a plausible actin network structure coupled with the unbalanced growth, which might account for the rotations of the beads predicted by the kinematic model. The findings of this study heighten the demand for further experimental work to verify the model we put forth by direct imaging of bead rotations, as well as for better mechanistic models of the bead rotations.