Redundant Mechanisms for Stable Cell Locomotion Revealed by Minimal Models

Abstract

Crawling of eukaryotic cells on flat surfaces is underlain by the protrusion of the actin network, the contractile activity of myosin II motors, and graded adhesion to the substrate regulated by complex biochemical networks. Some crawling cells, such as fish keratocytes, maintain a roughly constant shape and velocity. Here we use moving-boundary simulations to explore four different minimal mechanisms for cell locomotion: 1), a biophysical model for myosin contraction-driven motility; 2), a G-actin transport-limited motility model; 3), a simple model for Rac/Rho-regulated motility; and 4), a model that assumes that microtubule-based transport of vesicles to the leading edge limits the rate of protrusion. We show that all of these models, alone or in combination, are sufficient to produce half-moon steady shapes and movements that are characteristic of keratocytes, suggesting that these mechanisms may serve redundant and complementary roles in driving cell motility. Moving-boundary simulations demonstrate local and global stability of the motile cell shapes and make testable predictions regarding the dependence of shape and speed on mechanical and biochemical parameters. The models shed light on the roles of membrane-mediated area conservation and the coupling of mechanical and biochemical mechanisms in stabilizing motile cells.

Introduction

Eukaryotic cells crawl by making protrusions, contracting their cytoskeletons, and adhering to the surrounding environment in a diverse, complexly controlled and integrated sequence of events (1). Biophysical and biochemical processes combine to produce motile cells that can track down pathogens, determine organism development, repair wounds, and allow cancerous cells to metastasize (2). Diverse experimental research using biochemistry, microscopy, genetics, and biophysics has produced a wealth of data that describe the molecular pathways of cell migration, the interconnectivity of the networks involved in transport and turnover of the cytoskeleton, and the forces and flows that are produced inside the cell (3). Yet, our understanding of how these processes unite to produce a crawling cell is still incomplete. One major missing link is comprehensive quantitative models that can predict the shape, speed, and intracellular processes of a moving cell (4).

Here we focus on the best-understood process, lamellipodial motility of cells on flat surfaces (5,6), and do not discuss other, equally important, modes of locomotion (1,7). We address the question of how motile cells maintain their shape and speed, the significance of which is underscored by the fact that cell shape reflects various dynamic cellular processes, such as remodeling of the cytoskeleton underlined by biochemical signaling (8). Roughly speaking, the question about cell shape and speed breaks into the following queries: How does the rear retract to keep up with the protruding front? How are the sides contained from spreading and collapsing (Fig. 1)? We can best address these questions by considering rapidly and steadily crawling simple-shaped cells such as fish epithelial keratocytes. When single cells are placed on a flat surface, they assume a stereotypical half-moon shape with a broad, flat, motile appendage, the lamellipodium, and maintain nearly constant cell shape, speed, and direction (Fig. 1) over many cell lengths (5,6). Lee et al. (9) proposed a geometric principle for lamellipodial shaping in motile keratocytes whereby the cell boundary expands at the front and retracts at the rear in a locally normal direction with spatially graded rates, so that the advancement at the front is the fastest, and then smoothly decreases toward the sides. A simple trigonometric formula can be used to determine cell shape as a function of the expansion/retraction rates, but the mechanics and biochemistry behind this cell shape remain to be determined.

Figure 1.

Schematic illustrations of the four cell motility models examined in this work. (A) G-actin transport to the leading edge creates graded protrusion of the F-actin cytoskeleton. (B) Delivery of new cell membrane via microtubule-assisted vesicle transport controls the rate of protrusion at the leading edge. (C) The RhoGTPases Rac and Rho regulate protrusion (Rac) and contraction (Rho) within the cell. (D) Myosin binds to and contracts the actin cytoskeleton, creating cytoskeletal flows that redistribute the bound myosin.

In this work, we use moving-boundary simulations to model cell-shape dynamics. There is a rich history of investigators using such models to reproduce cell shape. Conceptually, one of the simplest models is one in which local stimulation and global inhibition of protrusive activity govern the local boundary velocity (10). Stéphanou et al. (11) added specific biophysics to the general model of Satulovsky et al. (10) by suggesting that membrane protrusions are induced by hydrostatic pressure and are opposed by tension from the actin filaments linked to the membrane and actomyosin contractility. The multiscale model of Rubinstein et al. (12) solved partial differential equations to describe the mechanics of an elastic actomyosin shell on the free boundary domain and reproduced the half-moon keratocyte shape. Another model considered the complex dynamics of two types of actin networks and reproduced the keratocyte shape by using moving-boundary simulations (13). The three-dimensional computations of Herant and Dembo (14) employed reactive interpenetrating flow theory, which considers the actin polymer network to be a very viscous fluid that moves through the fluid cytoplasm. The role of biochemical signaling in cell shaping was highlighted by Marée et al. (15), who demonstrated that the reactions and diffusion of Rho GTPases can provide robust maintenance of a steady, rapidly motile keratocyte shape. Another model, similar in spirit to that of Marée et al. (15), was also shown to reproduce keratocyte shapes (16).

All of these studies were very useful in elucidating plausible motile cell dynamics; however, they used different techniques, considered only a single mechanism of lamellipodial motility, and were either schematic (10,11) or very complex and elaborate (12,14–16). There are many indications that redundant mechanisms underlie cell motility (3,8,17), so we set out to examine whether different mechanisms can maintain similar shapes and movements, and whether they can complement each other in this task. We did this within a unified and conceptually simple framework of a level set-based, finite-volume method (18). Based on the introductory notes above, we investigated the dynamics, shape, and speed of cells that result from four simple models for cell motility.

We begin with a G-actin transport-limited model based on the idea that actin monomer depletion leads to graded G-actin concentration and therefore to graded F-actin growth. We test whether this mechanism can define the cell shape. Then, we investigate whether delivery of membrane components via microtubules as a part of a polarized endo/exocytotic cycle (19) can account for the stable motile cell shape. Third, we examine a hypothesis that feedback between two RhoGTPases, Rac and Rho, leads to a mutually exclusive assembly of two different kinds of dynamic cytoskeletal structures, which in turn promotes the establishment of a front–back polarity and stable cell locomotion. Fourth, we quantitatively test the hypothesis that myosin contraction of the actin network is the predominant mechanism for cell motility, i.e., that myosin molecules are swept to the rear of the cell and generate a centripetal actin flow that pulls the rear forward and contains the sides, thereby shaping the lamellipodium. We also study the roles played by membrane area constraint and front–back polarity in shaping the cell. Finally, we test whether some of the four proposed mechanisms can complement each other in stabilizing cell shape. (Further details regarding the motivation for these models are presented in the Supporting Material.)

Below, we report that numerical simulations of the models demonstrate that the G-actin, microtubule, and Rac/Rho models can reproduce stable lamellipodial shape and movement. On the other hand, by itself, the model that treats myosin contraction of the actin network as the predominant mechanism for cell motility cannot reproduce cell shape robustly, but together with either a graded actin treadmill or the Rac/Rho-regulation model, it can capture many features of keratocyte motility. Therefore, we propose that some of the underlying complexities of the biochemical and biophysical mechanisms of cell motility can provide the necessary redundancy for robust cellular migration.

Materials and Methods

In this section, we provide a qualitative description of the four motility models examined in this work. The mathematical details of the models are provided in the Supporting Material.

G-actin transport model

We start from one of the simplest possible mechanisms that can explain the shape and speed of a crawling cell: diffusion of G-actin from the rear, where F-actin disassembles, to the front and sides, where F-actin assembles (20) (Fig. 1 A). The idea that this is the limiting step in motility was previously explored in a limited and simplified form, in one dimension (21), but has not been proposed before to explain two-dimensional cell shape. To explore the crawling behavior predicted by this hypothesis, we constructed a model in which G-actin diffuses through the cell with diffusion coefficient D. We assume that the cell is polarized before motility occurs, i.e., a biochemical process inside the cell (e.g., as in the Rac/Rho model described below) activates polymerization at the leading edge of the cell and inhibits polymerization away from this region. Along the leading edge, F-actin grows at the rate k_ong(x) (21) in proportion to the local G-actin concentration g(x), where k_on is the polymerization constant. In the frame of the moving cell, the F-actin flows to the rear at the cell front and inwardly at the cell sides (12,22,23), and disassembles around the focal point to which the flow converges, largely in the region of the cell body, which is observed to be at the center of the cell rear. To account for this behavior, we define a circular region at the rear of the cell where F-actin disassembles at a constant rate, releasing monomeric G-actin, which then diffuses back to the leading edge. For simplicity, we do not consider nucleotide exchange on G-actin and reactions with thymosin and cofilin (24). The circular depolymerization zone moves with the cell rear, being pulled forward by the retracting cell membrane. We define the zone at the rear of the cell where actin does not polymerize in an ad hoc way, introducing a wide ellipse that is centered at the same location as the depolymerization zone. The location of that center moves with the center of the depolymerization zone. Any boundary points that fall within the ellipse have a polymerization rate equal to zero; all other boundary points expand with the rate αk_ong(x). In addition, we assume that the cell has a preferred area A₀ and that deviations from this area produce an effective restoring velocity that is locally normal to the cell boundary. The rationale for this assumption stems from the hypothesis (25) that the plasma membrane is stretched tight around the lamellipodium. Thus, growing actin filaments at the leading edge push the membrane forward and to the sides, generating membrane tension. This tension retracts the membrane and F-actin remnants at the rear, keeping the area constant, as observed previously (25).

Microtubule-associated vesicle transport model

The polymerizing actin network can grow forward against the membrane in several different ways. For example, excess membrane can be preserved in folds, or the membrane envelope can slide forward effortlessly in synchrony with the cell crawling (26). In both cases, the lamellipodial area can be preserved such that any advance of the leading edge will be strongly coupled to retraction of the rear (25). In the model described above, we tested the latter assumption. Here, we assume that the membrane hinders the advance of the actin network, and, in accordance with data from Howes et al. (27), that exocytosis of vesicles transported to the leading edge creates excess space that allows polymerization to protrude the leading edge. We also assume that the membrane is simultaneously removed at the rear. The membrane tension-mediated feedback assumed in the model ensures that the total plasma membrane area is conserved, so the membrane area inserted into the leading edge is immediately internalized at the rear. The cell rear is effectively hauled forward by recycling the membrane from the rear. Thus, we assume that the flux of membrane vesicles to the leading edge is rate-limiting for the protrusion and is set by the density of the microtubule ends that are in close proximity to the membrane (Fig. 1 B). To the best of our knowledge, a quantitative model such as this has not been proposed before. Microtubules originate at the microtubule organizing center (MTOC) and grow radially outward. Because the MTOC is located in front of the nucleus, microtubules are partially occluded from the rear of the cell, as indicated by many previous observations (28). Therefore, the position of the MTOC with respect to the nucleus naturally defines a polarity for the cell. Thus, the polarity mechanism, as in the previous model, is upstream of the motility and shaping mechanism examined here. The nature of the polarity mechanism is largely unknown; however, some plausible hypotheses are discussed elsewhere (28). The density of the microtubule ends that reach the cell periphery depends on the kinetic parameters of the microtubule dynamics and the radial distance from the MTOC to the cell periphery. We assume that the rate of extension along the edge is proportional to the flux of membrane vesicles, which in turn is proportional to the local microtubule end density at the edge.

Rac/Rho model

The Rho GTPases Cdc42, Rac, and Rho are strongly implicated in determining cell polarity and regulating cell motility. Activation of Rac leads to the formation of lamellipodia and membrane ruffles (29), and Rac has been shown to be upstream of actin polymerization (30). On the other hand, Rho activation produces stress fibers and focal adhesions (31), and Rho is known to act upstream of myosin light chain kinase, which induces myosin contraction of the actin network (32). Therefore, a possible model for cell polarity and motility is one in which Rac activation leads to polymerization at the leading edge of the cell, and Rho activation at the rear of the cell induces myosin contraction of the cytoskeleton, which hauls the rear of the cell forward (Fig. 1 C). We simulate this mechanism using a simple reaction-diffusion model that was previously proposed for Rho GTPase dynamics (33). In this model, the activated GTPase positively regulates its own production, whereas deactivation occurs at a basal rate. Because the activated protein binds to the membrane, diffusion of the active form is two orders of magnitude slower than the diffusion of the cytosolic inactive form. We use this model to describe the dynamics of Rac and Rho and their effect on cell motility. Because this model is known to spontaneously polarize for one-dimensional fixed geometries, it is possible that the dynamics of these proteins can lead to stable, directed migration. We assume that active Rac induces actin polymerization. Active Rho induces myosin contraction, and we assume that the active form of Rho mirrors activated Rac. This model is conceptually similar to but mathematically simpler than that proposed by Marée et al. (15).

Actomyosin contraction model

Finally, we consider a mathematically more complex model in which actomyosin contraction of the cytoskeleton aids actin polymerization as the predominant mechanism underlying motility (Fig. 1 D). Measurements of the response of the actin cytoskeleton in living cells show that applied force causes the actin network to flow like a fluid on timescales longer than a few seconds (34). Therefore, we treat the cytoskeleton as a viscous fluid with viscosity η. We assume that myosin molecules can bind and unbind from the actin, with rate constants K_on and K_off, respectively. While it is bound, myosin is assumed to exert an isotropic, contractile stress on the actin network that is proportional to the local concentration of bound myosin. This leads to a flow of the cytoskeleton that pulls the bound myosin with it. Unbound myosin rapidly diffuses and is therefore treated as being uniform throughout the cell. We model the interaction between the cytoskeleton and the substrate as a resistive drag force that is proportional to the velocity of the cytoskeleton. F-actin polymerizes at the cell boundary. In keratocytes, the myosin at the rear of the cell acts to bundle the actin, which aligns parallel with the cell membrane (20,35). Therefore, there are fewer barbed ends in contact with the membrane, and the polymerization rate is decreased. We model this by assuming that the polymerization rate at the cell membrane decreases with increasing myosin concentration. Our model is similar to that of Barnhart et al. (23); however, the latter was not simulated fully with a moving boundary. We assume that the cell area is restored to the preferred one by an internal pressure that works to preserve a roughly constant cell volume. We also considered two extensions of this model: First, we added graded actin protrusion to the actomyosin contraction model, in similarity to previously proposed hypotheses (23). Specifically, we redefined the boundary velocity of the actomyosin contraction model by making the constant actin polymerization rate a function of the angular coordinate θ (from the cell center-of-mass): V_p(θ) = 1 + β cos θ, where β is a constant. Second, we assumed that the actin protrusion rate is proportional to the concentration of active Rac, and the myosin stress is proportional to the concentration of active Rho.

Results

For all of the models considered here, we studied the dynamics of an initially circular cell driven by the prescribed mechanisms. We then investigated the dependence of cell shape and speed on the parameters of the different models. The reader can best appreciate the evolution of the stable motile cell shapes by viewing the movies in the Supporting Material.

Robust motile cell shape can be stabilized by the G-actin transport mechanism

The G-actin transport model depends on three different parameters: the assembly rate constant at the leading edge, the disassembly rate of the F-actin, and the diffusion coefficient of the G-actin. The diffusion coefficient can be scaled out of the problem, leaving just the dimensionless assembly and disassembly rates as free parameters. We simulate an initially circular-shaped cell with a uniform concentration of G-actin. The G-actin concentration is then depleted at the front of the cell and grows in the depolymerization zone. Points on the leading edge of the cell that are closer to the depolymerization zone receive a larger flux of G-actin, and thus polymerization is faster at these locations. At early times, faster polymerization occurs at the sides of the cell, which are closer to the rear, and the cell widens. The cell rear is pulled forward due to the conservation of cell area. As the rear of the cell moves forward, the depolymerization zone moves closer to the front of the cell, and eventually the cell front gets closer to the rear than the sides. Thus, the G-actin concentration is higher at the front than at the sides, and the front protrudes faster than the sides (Fig. 2, A–C). The geometric principle of shaping (9) ensures that the stable lamellipodial shape is a half disc with wings that protrude rearward (Fig. 2 C). This qualitative shape and its quantitative aspect ratio are very similar to those of keratocyte cells. The shape is not sensitive to any of the model parameters.

Figure 2.

(A–C) Time series of the G-actin transport model, showing a time point shortly after the beginning of the simulation (A), an intermediate time point (B), and the stable crawling shape (C). The cell starts as a circle and contracts at the rear as the front moves forward. The steady-state shape is a half-disk with wings off the sides. This shape is nearly independent of the parameters. The color map shows the G-actin concentration in dimensionless units; arrows show the boundary velocity (Movie S1). (D–F) Time series of the microtubule-associated vesicle transport model. The cell starts as a circle and contracts at the rear as the front moves forward. The steady-state shape is fan-shaped, and this shape is nearly independent of the parameters. The color map shows the microtubule density; black arrows show the boundary velocity, and the green circle is the MTOC (Movie S2). (G–J) Time series of the Rac/Rho model. (G–I) Early dynamics show that the cell shrinks and then flips into the active state. (J) At long time periods, the cell takes on an elliptical/half-moon shape. The color map shows the concentration of species a; black arrows show the boundary velocity, and yellow arrows show the cytosolic velocity (Movie S3).

If F-actin disassembly increases, a larger pool of G-actin forms in the cell. Because the polymerization rate at the leading edge is proportional to the concentration of G-actin, the stable crawling speed of the cell is proportional to the magnitude of the source term (Fig. S1). Of interest, the crawling speed is nearly independent of the actin polymerization rate. This is because the rate at which the edge advances is proportional to the polymerization rate, but the pool of G-actin is depleted proportionally to the F-actin polymerization rate. These two effects compete against each other to make speed independent of the assembly rate. Cell speed in the G-actin transport model is proportional to the actin disassembly rate, which is along the lines of the effect observed by Cramer (36). These results regarding shape and speed are not counterintuitive; however, it is essential to obtain them by mathematics.

The microtubule-associated vesicle transport mechanism can also support a stable motile cell shape

Simulations of the microtubule-associated transport model produce a fan-shaped crawling cell (Fig. 2, D–F) characteristic of motile keratocytes. The shape and its aspect ratio are not strongly affected by the parameters. Note that the discontinuity in the boundary velocity at the corners of the resulting fan-like shape, which occurs at the point that microtubules cannot pass due to the cell body occlusion, is the cause of the sharp corners at the cell sides. The two main parameters of the model are an effective velocity that describes the membrane transport velocity multiplied by a probability factor for the membrane delivery, γ = V₀Nv-/2π(v₊ + v₋), and the magnitude of the effective membrane area-restoring term, B. Not surprisingly, the crawling speed of the cell is linear in γ (Fig. S2 A). For values of the area-restoring term B > 1, the speed is constant; however, the speed goes to zero with B when B < 1 (Fig. S2 B). Thus, area preservation is essential for this mechanism to work. If the rear is allowed to slack off, the cell boundary moves farther from the leading edge, membrane delivery slows down, and the protrusion decreases.

The Rac/Rho model predicts a stable motile cell shape without the area conservation term, but not with it

As observed in a previous study (33), we find that cells will spontaneously polarize; therefore, the Rac/Rho model does not require an upstream polarization mechanism. We considered an initial condition with a uniform concentration of inactive protein and a nearly constant concentration of active protein. If the active protein concentration is perturbed with a random initial condition, the direction of polarization will be random, as expected from previous results (33). We also considered an initial concentration that was perturbed with a small constant gradient. This initial cell state polarizes along the gradient direction and reaches steady state more quickly than when a random initial condition is used. The steady-state shape and speed are not affected by our choice of initial condition. Therefore, for most of our simulations, we perturbed the active protein concentration with a small uniform gradient.

We considered two separate scenarios: a cell with a preferred area and a cell with no preferred area. We found that when the cell area is not fixed, the cell initially begins to shrink (Fig. 2 G) because the concentration of active protein is below the threshold value. Therefore, myosin contraction driven by Rho exceeds the polymerization velocity produced by active Rac. As the cell shrinks, Rac at the front of the cell flips into the higher stable concentration, which causes the front of the cell to begin polymerizing faster than the myosin-induced contraction (Fig. 2, H and I). The cell eventually reaches a stable crawling shape with a high concentration of active Rac (low concentration of active Rho) at the front of the cell and a low concentration of active Rac (high concentration of active Rho) at the rear of the cell (Fig. 2 J). The velocity of the cell and the cell area are both proportional to the total protein concentration (Fig. S3); however, the cell shape is not strongly influenced by the total protein concentration, as the aspect ratio is relatively constant. It is somewhat unexpected that the aspect ratio of the cell is such that the cell is wider in the direction perpendicular to the direction of motion, because energetic considerations suggest that the system would evolve to a state that would minimize the length of the transition zone between the active and inactive states. However, in our system, the velocity of the boundary motion is proportional to the concentrations of active Rac and Rho. Because the polymerization rate is roughly constant on either side of the transition zone, the boundary dynamics favors a cell shape that is wider along the direction of the transition zone.

Of interest, if the cell area is fixed, the cell is no longer capable of achieving stable locomotion. As before, the cell still polarizes and begins to crawl. However, the rear of the cell eventually gets too close to the front of the cell, which causes the Rac at the rear of the cell to flip into the highly active stable state. The entire cell then has a uniform distribution of active Rac. The lack of polarization prevents further motion. We describe the implications of these results in the Discussion.

The actomyosin contraction mechanism alone does not stabilize the cell

Nondimensionalization of the actomyosin contraction model leads to four dimensionless parameters that influence the overall behavior of the model: the myosin-binding rate K_on and the myosin-off rate K_off, both of which are multiplied by the characteristic time L/V₀ (f = σ₀M_T/Ω, where σ₀M_T/L² is the characteristic magnitude of the myosin-generated stress, Ω = ζV₀L³ is the characteristic adhesion drag stress multiplied by the cell area, and ɛ = η/ζL² is the squared ratio of the characteristic length (η/ζ)^1/2 on which the actin flow is induced by local myosin contraction to the cell size). To account for the bundling of actin by myosin, we assumed that bundled actin polymerizes more slowly at the leading edge and defined a parameter α that determined the reduction of protrusion by myosin. We chose typical values of these parameters to be K_on = 0.25, K_off = 0.5, σ₀ = 0.2, η = 0.2, and α = 1, and investigated the behavior of the model near these typical values. We used an initial condition with a nearly uniform distribution of bound myosin that was perturbed by a small constant gradient between the front and back of the cell. We found three different behaviors for the evolution of the cell morphology. First, for small values of the myosin contractile stress (f ≤ 1), the cell remains circular and there is flow of actin toward the center of the cell (Fig. 3 A), in agreement with previous observations of keratocyte fragments (37,38). For larger values of the myosin contractile stress, F-actin is rapidly pulled into the center of the cell, which produces a large concentration of myosin at the center. The small asymmetry in the cell shape that is produced by the initial conditions causes the cell to pinch in the center (Fig. 3 B). The third morphology arises when the viscosity is small compared with the substrate drag (ɛ << 1) and myosin strongly inhibits protrusion at the cell membrane. In this situation, the substrate drag dominates and the effect from myosin is highly localized. For large initial gradients in the myosin concentration, it is possible to obtain a contractile velocity that can pull the rear forward. The initial stages of the simulation look fairly good, with the cell becoming somewhat half-moon-shaped. However, polymerization away from the high myosin concentration continues to spread the cell out, and the cell becomes very wide and thin (Fig. 3 C).

Figure 3.

Evolution of the actomyosin contraction model for three different sets of parameters. The color maps show the concentration of bound myosin (blue, low concentration; red, high concentration) and the arrows show the actin flow field. (A) When myosin stress is small, the cell remains circular and the myosin is swept to the center of the cell by a centripetal flow of actin. (B) At larger values of the myosin stress coefficient, the cell pinches at the center. (C) When myosin inhibits protrusion by bundling actin, and the actin viscosity is low, the cell initially takes on a keratocyte-like appearance with a high concentration of myosin at the rear and a half-moon shape. However, this shape does not persist and the cell extends sideways and gets very thin. Simulations use the parameter values listed in the text, except for (A) η = 0.3; (B) η = 0.1; and (C) α = 4, η = 0.05, and σ₀ = 0.1 (Movie S4, Movie S5, and Movie S6).

These results are intriguing because the model seems to accurately describe the behavior of a keratocyte fragment before the symmetry breaks, in which case the actin cytoskeleton and presumably other factors that influence motility remain fairly isotropic. However, this simple model is incapable of producing a steadily moving cell, which suggests that some other factors are missing. In the following sections, we consider two possible additions to the model that can produce steady crawling: 1), incorporating a graded actin treadmill in which cell polarity is defined by an asymmetric protrusion rate; and 2), using the previously described Rac/Rho model to regulate the actomyosin contraction model.

The actomyosin contraction mechanism stabilizes the motile cell when it is coupled with the graded actin treadmill

In recent experiments, Barnhart et al. (23) examined keratocyte motility on substrates where the adhesive strength was modified, and reported that at low adhesion, the actin polymerization rate appeared to be high at the front of the cell and low at the rear, for unknown reasons. We decided to explore whether adding graded actin protrusion to the actomyosin contraction model could produce keratocyte-like motility in the regime where parameters f ≤ 1and ɛ ∼1. We redefined the boundary velocity to be ${\mathbf{v}}_{\text{b}}=\left({\text{V}}_{\text{p}}\left(\theta \right)/\left(1+\alpha \text{m}\right)+\mathbf{v}·\mathbf{n}\right)\mathbf{n}$, where the protrusion velocity is V_p(θ) = 1 + β cosθ, as indicated in the model description.

We first considered the case in which α = 0 (myosin does not affect actin polymerization). If the myosin stress is also equal to zero, the cell moves as a disk. Constant polymerization of actin at the cell edge produces centripetal flow of actin into the center of the cell; however, drag between the actin and the substrate slows the flow of actin with respect to the cell motion, and the bound myosin is swept to a location that lags behind the cell center (Fig. 4 A). Increasing the myosin stress causes the back of the cell to cave in slightly (Fig. 4 B) and also speeds up the cell motion (Fig. 4 C). Myosin-mediated acceleration of the cell is in agreement with previously reported measurements (20). However, purely increasing the myosin stress term is insufficient to produce a realistic half-moon shape. We added back in the tendency for myosin to bundle the actin and thereby reduce the protrusion rate by increasing the value of α. The reduction in protrusion at higher concentrations of myosin leads to reduced protrusion of the actin at the rear of the cell. Therefore, the myosin is not swept away from the rear, and accumulates there. It is then possible to find parameter regimes that produce steadily translocating, half-moon shapes with realistic actin flows (6,22,23) and myosin distribution (Fig. 4 D). The stable realistic motile cell shape evolves if the scaling of the actin network viscosity, adhesion strength, and myosin stress is such that f ∼1 and ɛ ∼1. First, this means that the characteristic distance (η/ζ)^1/2 on which mechanical stress spreads in the cell is of the order of the cell size, which parallels recent modeling and measurements of the actomyosin contraction in the cell cortex (39). Second, this means that the myosin stress resisted by adhesion generates actin flow at the rear that has to be comparable to the actin polymerization rate at the front.

Figure 4.

(A–D) Actomyosin model with a graded actin treadmill. (A and B) When actin protrusion is independent of myosin concentration, the cell maintains a nearly circular shape. Increasing the myosin stress coefficient leads to a small buckling of the cell rear that increases with increased stress. Simulations use typical parameter values, with values of β = 0.5 and σ₀ = 0.02 (A) and 0.3 (B). (C) The stress from myosin also causes the cell to crawl faster. (D) When myosin alters actin protrusion, it is possible to obtain a steadily translating, realistic half-moon shape, with actin flows and myosin localization similar to those observed in experiments. Parameter values: β = 0.7, α = 0.3, η = 0.2, k_off = 0.2, k_on = 0.2, and σ₀ = 0.05. The color map shows the myosin concentration, and green arrows show the actin flow field (Movie S7, Movie S8, and Movie S9). (E and F) Combined actomyosin contraction and Rac/Rho model. For some parameter values, the cell achieves a steady, crawling half-moon shape. In this simulation, α = 1, η = 0.1, k_off = 0.5, k_on = 0.5, and σ₀ = 0.1. The color map shows the concentrations of active Rac (E) and bound myosin (F), and the arrows show the actin flow field (Movie S10).

The actomyosin contraction mechanism also stabilizes the motile cell when coupled with the Rac/Rho model

The results presented above suggest that the myosin contraction model can work to produce steady, crawling keratocyte shapes as long as an additional polarization mechanism is present. The actin treadmill is an imposed polarization. One possible self-polarizing mechanism is the spontaneous polarization of Rac and Rho. Therefore, we put the actomyosin system under the control of Rac and Rho model that we examined earlier. As before, we assumed high concentrations of active Rac corresponding to low concentrations of Rho. We then assumed that the protrusion rate of actin was proportional to the concentration of active Rac (i.e., the protrusion rate is [Rac]/(1 + αm)), and the myosin stress was proportional to the concentration of active Rho (i.e., the myosin stress is σ₀[Rho]m). We found that, for a range of values around the typical parameter values of the actomyosin contraction model, regulation by the simple Rac/Rho model was sufficient to produce a steadily translocating, half-moon-shaped cell (Fig. 4, E and F). In these simulations, a small perturbation in the initial conditions produced a high concentration of active Rac at the front of the cell and consequently a high concentration of active Rho at the rear. The cell began to crawl by stretching out in the direction of motion and becoming concave at the rear. As the cell continued to crawl, it slowly widened and became more half-moon-shaped. The crawling persisted for a long time but slowed down somewhat because the widening of the cell brought the rear closer to the front, where a high concentration of active Rac was found.

Parameter values significantly different from the default ones did not lead to stable, persistent motion. In all of these cases, morphological changes caused the Rac concentration to become uniform in either the active or inactive state. When this occurred, a radial flow of actin eventually was produced that swept the myosin to the center of the cell, and the cell stopped moving. Sometimes this would happen quickly, and the cell would not move much at all over the course of a simulation. In other simulations, the cell would crawl for a period of time, but then the Rac concentration would flip into one of the two stable states and the cell would then stop moving (Fig. S4).

We also found situations in which the cell would crawl steadily, but there were periodic oscillations at the leading edge of the cell. These were observed when the myosin on and off rates were large and the viscosity was low. Periodic oscillations in motile keratocytes were reported previously (23), and it is not out of the question that our model provides an explanation for these observations.

Discussion

Previous studies provided rigorous proof of the existence of motile cell-like shapes in characteristic cell motility models (40); however, the local and global stabilities of such stationary motile shapes have not been investigated systematically. Here, we show numerically that a wide class of motile models can reproduce stable cell shapes, and that the level set method is a robust tool for simulating the dynamics of motile cells. In the future, as quantitative details of the molecular mechanisms that govern cell migration become clear, it will be easy to couple more mathematical models with the existent ones and rigorously explore the resulting dynamics. We plan to implement the codes of this work within the user-friendly framework of the Virtual Cell (41), so that investigators in the modeling and biology communities will be able to test which combinations of motile models drive the migration of specific cell types.

The main findings of this work are summarized in Table S1 for the reader's convenience. Our main goals in this study were to introduce conceptually novel models of cell shaping by G-actin and microtubule-based transport, and to investigate whether existent models of actin-myosin contraction and Rac/Rho-regulation can predict stable two-dimensional cell shape, separately or in combination. We have demonstrated that three simple mechanisms—spatial grading of the protrusion-limiting G-actin concentration due to diffusion, negative feedback between Rac and Rho resulting in the mutually exclusive assembly of two different kinds of dynamic cytoskeletal structures, and delivery of membrane components via microtubule-limiting protrusion—can reproduce stable lamellipodial shape and movement. The aspect ratio of the characteristic half-moon shape is remarkably insensitive to the model parameters and is of the same order of magnitude as that observed in motile keratocyte cells (25). We have shown that a model in which myosin contracts the actin cytoskeleton can by itself predict globally unstable shapes. This is a novel (to our knowledge) and interesting conclusion, because the idea that actomyosin contraction can be the principal mechanism for cell shaping was suggested previously (19,20) and qualitatively makes perfect sense. Previous numerical simulations on fixed cell shapes (21,22) cast doubt on the ability of this mechanism alone to stabilize the cell shape, but this issue was never investigated fully and consistently. However, we also found that when this model is coupled with either a graded actin treadmill or Rac/Rho regulation, it can predict robust and stable keratocyte-like motility.

Which of the four simulated models are actually relevant to keratocyte motility? It is known that microtubules are not necessary for keratocyte movements (42), but the possibility that biochemical regulation models play a role cannot be definitively ruled out. Of interest, our simple biochemical regulation model does not work properly with the area-preservation term, whereas in keratocytes the area preservation due to membrane inextensibility seems to be important (25). All other models considered in this study work well with, and in fact require, area preservation terms. There is an experimental hint that the G-actin concentration by itself is not the key factor in regulating motility, because the cytoplasm simply contains too much actin in monomeric form (43). Rather, partitioning of the G-actin in a number of forms marked by the hydrolysis state and actin-binding proteins can, in addition to diffusion, regulate motility. Finally, quantitative data (23) suggest that the actomyosin contraction model complements the graded actin treadmill in shaping the motile cell.

The important question about keratocyte polarization and motility initiation remains open. The available data (37,38) suggest that myosin and perhaps Rac/Rho activity are essential for the observed spontaneous symmetry breakage of the disc-like stationary cell and evolution of the motile half-moon lamellipodium. Of interest, in agreement with those observations, our results suggest that the myosin contraction and Rac/Rho regulation models have properties of both the stationary, nonmotile state (i.e., they are unstable) and the polarized motile state (i.e., they evolve spontaneously). On the other hand, the G-actin and microtubule transport models need additional mechanisms to impose front–back polarization for steady motility to occur.

In an insightful qualitative analysis, Lämmermann and Sixt (7) pointed out that various spatial-temporal combinations of three processes—protrusion, contraction, and adhesion—can be used by the cell to adapt to any two- or three-dimensional environment and to produce any mode of locomotion that has been observed. To do so, the cell has to deploy various combinations of redundant motility models. Here we considered four such models and showed that they can shape and stabilize the motile cell separately or in combination. In this study we did not consider additional models, such as the dynamic graded adhesion treadmill, that have only recently begun to be quantified (44). We plan to add such models to our repertoire, after which quantitative examination of the motile machinery in its entirety will be within reach.