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The authors have declared that no competing interests exist.

Conceived and designed the experiments: JZ AM. Performed the experiments: JZ AM. Analyzed the data: JZ AM. Contributed reagents/materials/analysis tools: JZ AM. Wrote the paper: JZ AM.

Two theoretical models dominate current understanding of actin-based propulsion: microscopic polymerization ratchet model predicts that growing and writhing actin filaments generate forces and movements, while macroscopic elastic propulsion model suggests that deformation and stress of growing actin gel are responsible for the propulsion. We examine both experimentally and computationally the 2D movement of ellipsoidal beads propelled by actin tails and show that neither of the two models can explain the observed bistability of the orientation of the beads. To explain the data, we develop a 2D hybrid mesoscopic model by reconciling these two models such that individual actin filaments undergoing nucleation, elongation, attachment, detachment and capping are embedded into the boundary of a node-spring viscoelastic network representing the macroscopic actin gel. Stochastic simulations of this ‘in silico’ actin network show that the combined effects of the macroscopic elastic deformation and microscopic ratchets can explain the observed bistable orientation of the actin-propelled ellipsoidal beads. To test the theory further, we analyze observed distribution of the curvatures of the trajectories and show that the hybrid model's predictions fit the data. Finally, we demonstrate that the model can explain both concave-up and concave-down force-velocity relations for growing actin networks depending on the characteristic time scale and network recoil. To summarize, we propose that both microscopic polymerization ratchets and macroscopic stresses of the deformable actin network are responsible for the force and movement generation.

There are two major ideas about how actin networks generate force against an obstacle: one is that the force comes directly from the elongation and bending of individual actin filaments against the surface of the obstacle; the other is that a growing actin gel can build up stress around the obstacle to squeeze it forward. Neither of the two models can explain why actin-propelled ellipsoidal beads move with equal bias toward long- and short-axes. We propose a hybrid model by combining those two ideas so that individual actin filaments are embedded into the boundary of a deformable actin gel. Simulations of this model show that the combined effects of pushing from individual filaments and squeezing from the actin network explain the observed bi-orientation of ellipsoidal beads as well as the curvature of trajectories of spherical beads and the force-velocity relation of actin networks.

Cell migration is a fundamental phenomenon underlying wound healing and morphogenesis

Here we examine computationally the mechanics of growing actin networks. This problem has a long history starting from applying thermodynamics to understand the origin of a single filament's polymerization force

(A) Tethered ratchet model. Actin filaments (gray) can attach to the obstacle surface (black line) via attachment sites (yellow) and exert pulling forces (

In parallel to these microscopic theories, macroscopic elastic propulsion model

Note that the widely used terminology could be confusing as the elastic propulsion theory is sometimes called mesoscopic rather than macroscopic. Both terms are justified: the macroscopic mechanics is described using continuum theory, but an actin layer of a few microns thin is certainly a mesoscopic system. The model we present is mesoscopic in the sense that it spans from the microscopic level of individual filaments to the macroscopic level of continuous description of an actin gel. The model is also hybrid because it takes into account both local discrete forces and global network stress. We will mostly use the term “hybrid” throughout the paper.

The first simple attempt to use hybrid modeling of the lamellipodial edge was recently made in

Besides the force-velocity relation, the non-zero curvatures of the trajectories of motile objects

Below, we describe observations of ellipsoidal, rather than spherical, beads that cannot be explained by either microscopic or macroscopic model. This, as well as the complex force-velocity relation and curvature distribution described above, hints that perhaps a hybrid model with individual actin filaments pushing from the surface of a macroscopic deformable actin gel can explain the experiments better. Recent experiments and theory

We developed a two-dimensional (2D) simplification of a 3D hybrid model (

Recently, with our experimental collaborators, we reported observations of the ellipsoidal beads that were uniformly coated with an actin assembly-inducing protein (ActA)

(A–B) Fluorescent images show actin tails of the motile beads. The dark ellipsoidal shapes at the fronts of the tails illustrate bead's propulsion along its (A) long-axis and (B) short-axis. The detailed statistics of phase contrast images reported in

To see whether the two existing models of actin propulsion can explain this result, we simulated the motion of actin-propelled ellipsoidal beads as described in the

Thus, the elastic propulsion model predicts that beads only move along their long axes, while microscopic ratchet model predicts that beads only move along their short axes, and neither model can explain the observation. In contrast, the full hybrid model predicts that the bead can move in both orientations due to the combination of the elastic squeezing and the geometric spreading of actin and switch infrequently between them (Video S1,

The simulation results of the effects of a bead's aspect ratio (at constant area of the bead) on its orientation are shown in

We vary the Young's modulus of the actin network by varying the spring constant in our model as described in

The effect of the ratio of the number of attached to the number of pushing filaments,

To further test the hybrid model, we simulated the motion of actin-propelled spherical beads (

(A) Simulation snapshot of the hybrid model. Black circle: bead. White: actin network. Bar:

Indeed, the model predicts that when the detachment rate of actin filaments becomes low and a greater fraction of filaments is attached to the bead surface, beads start to have pulsatory motion due to temporary entrapment by the actin gel (

To obtain the distribution of the curvatures of the trajectories, we smoothed the simulated bead's trajectory to remove the high frequency noises and calculated (see

(A) Probability distribution of the normalized trajectory curvature for default values of parameters (open red circles), twice the value of attached to pushing filament ratio (green pluses) and twice the bead radius (dotted line), compared to a Gaussian distribution (solid black line). (B) Probability distribution of the normalized trajectory curvature with

When the detachment rate is low, we find that the curvature distribution becomes sharply peaked at zero (

We found that the predicted characteristic value of the root-mean-square curvature,

We also studied how the bead radius,

Two possible mechanisms may contribute to the turning of beads' trajectory: turning induced by elastic and ratchet torque, and turning induced by actin tail-reorientation (see

Note that in contrast to our results, a non-Gaussian distribution of the curvatures of trajectories of the beads was observed in

We simulated growth of an actin pedestal against flat elastic cantilever and force-clamped spherical bead, as in experiments

(A–B) Snapshots of hybrid model simulations. Blue: obstacles. White: actin networks. Dark gray: rigid substrate. Bars:

The simulated force-velocity relation predicted by the hybrid model for the flat cantilever is compared to the experimental data

We then used the hybrid model to simulate the force-velocity relation for the force-clamped bead. In this case, the force-velocity relation is concave-up, in good agreement with the observations

To investigate the effect of the filament attachments to the surface on the force-velocity relations, we varied the value of the attachment rate to change the ratio of the number of attached to the number of pushing filaments,

Complexity of the relation between geometry of the curved surface, molecular pathways of actin polymerization against this surface and resulting force

The elastic propulsion theory predicts that squeezing of the ellipsoidal beads orients them so that motility along the long axes ensues, while geometric effect of spreading of branching actin filaments results in beads moving along their short axes. Separately, the existing theories cannot explain the observed bi-orientation of the beads. Our hybrid model posits that the combination of the elastic squeezing and geometric spreading leads to bi-orientation and reversible switching between two orientations, in agreement with the observations. To test the hybrid theory in the future, we propose to vary the bead geometry and concentrations of actin accessory proteins, thus modulating the network stiffness and interactions with the surface. Our model makes specific, nontrivial and testable predictions (see

The hybrid model reproduces the observed order of magnitude of curvatures of the trajectories in 2D and suggests that switching between the low- and high-curvature trajectories is caused by the temporary entrapment of the beads in the actin gel. The model predicts a Gaussian distribution of the curvatures for fast-moving beads due to random fluctuations of filament numbers and redistribution of actin around the bead's surface. In agreement with observations, our simulations show an additional sharp peak at zero curvature in the curvature distribution for slowly-moving beads. Importantly, the model suggests that elastic effects have little impact on the distribution of trajectory curvatures for fast-moving beads, while for beads that tend to be trapped in the actin cloud due to frequent filament attachments, the elastic effects are responsible for deviations from Gaussian distributions.

The hybrid model posits that the qualitative difference between two force-velocity measurements

In the present form, our model has a number of limitations. The main one is that due to computational time limitations, we simulated the model in 2D as a simplification of a 3D system. So, rigorously speaking, all our results are applicable to cylindrical, rather than spherical objects. In Ref.

Due to these limitations, our model does not capture some observed effects. Notably, the simulations do not reproduce observed hysteresis in the growth velocity of actin networks under force

Another open question is relation of our model to other theories of the actin-based propulsion. Those include microscopic models of propulsion by tethered actin filaments

Motility experiments on ellipsoidal beads were carried out in the lab of J. Theriot as previously described

Spherical beads were prepared in the lab of J. Theriot as previously described

For both experiments, positions and orientations of beads were computed from phase-contrast images and assembled into tracks as described in

In the hybrid model (

Once capped, the filament is removed from the simulation, since in reality it will stop growing and cannot attach to the surface to exert pulling forces. However, the node corresponding to the pointed end of the filament remains, so this filament effectively becomes a part of the deformable network. We do not track the orientation of individual pushing filaments, but treat them as coarse-grained clusters of actual filaments that always push perpendicularly to the obstacle surface (see

The deformation of the network is represented by the motion of nodes and springs in the network, which is obtained by moving all the nodes toward their force-equilibrium positions at each time step. For actin-propelled beads, we assume that the nodes in the network become immobile when they are more than a few microns away from the bead surface, representing the adhesion of the actin tail to the substrate. The bead moves and rotates to satisfy the force and torque balances from the filaments. For the force-velocity measurements, we fix the network at the bottom and allow all the rest nodes to move to reach force balance. The network undergoes disassembly, which is treated by removing the nodes and their connected springs from the network randomly with a rate proportional to the number of existing nodes. We have also included the effect of rupture of crosslinks by introducing a critical stretching force, above which the links break and get removed from the network. During the steady motion of beads, the creation and extinction rates of actin networks balance, causing a treadmilling actin tail behind the bead (Video S1). Effective viscoelastic behavior of the actin network emerges from the disassembly and breaking of the network. Further details about the model equations and parameters are described in

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The authors are deeply grateful for experimental data shared by Julie Theriot, Lisa Cameron, Paula Giardini Soneral, and Catherine Lacayo. We also thank Catherine Lacayo and Julie Theriot for useful comments and help with the data analysis.