Modeling cell protrusion predicts how myosin II and actin turnover affect adhesion-based signaling

Abstract

Orchestration of cell migration is essential for development, tissue regeneration, and the immune response. This dynamic process integrates adhesion, signaling, and cytoskeletal subprocesses across spatial and temporal scales. In mesenchymal cells, adhesion complexes bound to extracellular matrix mediate both biochemical signal transduction and physical interaction with the F-actin cytoskeleton. Here, we present a mathematical model that offers insight into both aspects, considering spatiotemporal dynamics of nascent adhesions, active signaling molecules, mechanical clutching, actin treadmilling, and nonmuscle myosin II contractility. At the core of the model is a positive feedback loop, whereby adhesion-based signaling promotes generation of barbed ends at, and protrusion of, the cell’s leading edge, which in turn promotes formation and stabilization of nascent adhesions. The model predicts a switch-like transition and optimality of membrane protrusion, determined by the balance of actin polymerization and retrograde flow, with respect to extracellular matrix density. The model, together with new experimental measurements, explains how protrusion can be modulated by mechanical effects (nonmuscle myosin II contractility and adhesive bond stiffness) and F-actin turnover.

Significance

Cell migration is essential for development and homeostasis of tissues. To move, cells must balance the need to protrude the leading-edge membrane forward and to adhere to the surrounding extracellular matrix. This is inherently complex, because protrusion and adhesion do not operate in isolation. In fact, adhesion structures are coupled to the actin cytoskeleton (the protrusion machinery) through both direct and indirect mechanical effects. The indirect effects, mediated by locally activated, biochemical signaling pathways, are critical yet incompletely understood. In this work, a mathematical model integrating adhesion, signaling, and cytoskeletal dynamics at the leading edge was developed. Supported by quantitative experiments, analysis of the model unravels how the numerous interdependent effects influence cell protrusion.

Introduction

Orchestration of cell migration, directed by soluble and extracellular matrix (ECM)-associated factors, is essential for wound healing, embryonic development, and the immune response (1, 2, 3). Efficient cell migration involves the coupling of adhesion, signaling, and cytoskeletal subprocesses, which must be integrated across spatial and temporal scales (4, 5, 6, 7). In mesenchymal cells such as fibroblasts, which are chiefly responsible for wound repair, integrins engage ECM proteins and cluster to form adhesion complexes, which mediate both biochemical signal transduction and physical interaction with the F-actin cytoskeleton (5,8, 9, 10). Small nascent adhesions form at the cell’s leading edge and engage a network of branched F-actin that characterizes a protruding region of the cell (lamellipodium). As this structure advances, the stationary nascent adhesions either disassemble or grow to form larger focal adhesions (11). During their lifetime, nascent adhesions mediate activation of the small GTPase Rac and other signaling pathways that promote F-actin polymerization via nucleation of new barbed ends, driving membrane protrusion (12,13). While under mechanical tension, adhesions also promote activation of another small GTPase, RhoA, which initiates a pathway that stimulates nonmuscle myosin II (NMII) motor activity (14,15). NMII engages F-actin and causes contraction of the network, which works to increase its rearward (retrograde) flow and reduce membrane protrusion (16, 17, 18, 19, 20). Integrated with those signaling roles, nascent adhesions associated with F-actin constitute a mechanical clutch that resists retrograde flow, manifesting as traction force exerted upon the ECM/substratum (21, 22, 23).

Mathematical modeling has proved valuable in understanding isolated aspects of leading-edge motility (6,24,25). Starting from the classic conceptual model of dendritic actin treadmilling (26, 27, 28), Mogilner and Edelstein-Keshet formulated a partial differential equation (PDE) model of actin treadmilling and membrane protrusion opposed by the boundary stress, assuming a constant flux of new barbed-end generation at the membrane boundary and no retrograde flow of F-actin (29). Subsequent models added finer details of actin dynamics (30, 31, 32, 33, 34, 35, 36), while others characterized mechanical slippage of the F-actin network, clutched by adhesive bonds, that allows retrograde flow (34,37, 38, 39, 40). Other models have integrated NMII association with, and contraction of, the F-actin network to explain the intracellular forces that determine membrane protrusion, F-actin retrograde flow, and cell shape (41, 42, 43, 44). Together, these models successfully describe the minimal mechanical elements of leading-edge motility; however, they do not incorporate signal transduction, which couples integrin-mediated adhesion (and ligation of other receptors, as in chemotaxis) to the biochemical regulation of actin polymerization, F-actin turnover, and NMII motor activity.

In contrast to the modeling work cited above, other models have focused on signal transduction mechanisms, either hypothetical or based on known pathways, to explore control of leading-edge protrusion or cell polarization induced by a spatial cue (45, 46, 47, 48, 49, 50). Depending on the research question posed, these models either stop at the level of a key signaling output or describe interactions between signaling variables and a “toy” or phenomenological model of cytoskeleton/motility that typically ignores mechanics. They also vary with respect to spatial resolution. In the context of adhesion-based signaling, this is important because adhesion complexes may be considered stationary; as the protruding edge advances, existing adhesions are left behind. In fibroblasts, the estimated dynamic range of Rac-GTP (governed by the balance of diffusion and GTP hydrolysis rate) is only ∼2 μm (45,51); clearly, proximity matters.

In this work, we document the formulation and analysis of an integrative, spatial model incorporating adhesion, signaling, and actin dynamics, and mechanics affecting them, in lamellipodia. The model is deterministic and mechanistic, but it is coarse grained, such that the number of adjustable parameters is manageable. As we discuss, much more mechanistic detail could readily be added, with the burden of more parametric complexity. Even with an economical granularity, the model yields both intuitive and nonintuitive results, supported by previously published and new experimental measurements. One major insight is that adhesion-based signaling is localized proximal to the leading edge, stimulating F-actin polymerization, leading-edge protrusion, and nascent adhesion formation. Although we have considered this adhesion-based positive feedback loop previously (45,48,52), the present model explicitly connects Rac signaling to F-actin branch nucleation and generation of barbed ends that share mechanical load. Accordingly, there are two distinct, but interdependent, mechanical failure modes for the positive feedback and leading-edge protrusion: failure of the adhesion/F-actin clutch and of actin polymerization under load. Stress on the F-actin network applied by NMII contractility can contribute to both failure modes by destabilizing adhesions. These mechanical effects are influenced in complex ways by the compliance of the clutched adhesions, which is affected by substratum stiffness. Another major insight concerns the role of F-actin turnover, which is mediated largely by ADF/cofilin proteins (53). Although enhanced turnover reduces total F-actin available for stabilizing adhesions, as predicted previously (48), the present spatial model shows that the lost adhesions are farther away from the leading edge and therefore contribute less to pro-protrusion signaling than proximal adhesions. Moreover, by considering the availability of G-actin, the model predicts that the net effect of increased turnover is to enhance protrusion, mainly by increasing the flux of G-actin at the leading edge.

Materials and methods

Model formulation and implementation

We have constructed a spatiotemporal model of adhesion, cytoskeletal, and signaling dynamics governing protrusion of a lamellipod in a mesenchymal cell. The model is composed of PDEs and associated boundary conditions, describing reaction, diffusion, and advection of the various biochemical species. The model also incorporates a momentum balance on the F-actin network and the mechanics of clutched adhesions. A full description of the model equations and parameter values is provided in Text S1. The one-dimensional system, with 11 variables and 35 adjustable parameters, was solved in MATLAB (The MathWorks, Natick, MA) on a personal computer and using the North Carolina State University High-Performance Computing cluster.

A custom routine was written to solve the system of coupled PDEs with ordinary differential equations (ODEs) as boundary conditions. The pdepe function in MATLAB, which solves one-dimensional parabolic and elliptic PDEs, was used to solve the system of PDEs, while the ode15s function was used to solve the system of ODEs. Within each of the two functions, the time step is determined by the algorithm. A time loop was devised to sequentially solve the ODEs and PDEs. The ODEs were solved first for a small time step (δT = 0.5 s), and the results were fed into the coupled PDE solver for the same time interval. To calculate values for variables described by the ODEs, interpolation was used. This process was repeated until a steady state was reached. A spatial step (δx) of 0.05 μm was used. To test the dependence of this method on the choice of time and spatial step, the time step was modulated while keeping the ratio (δT/δx) constant (Fig. S3). This did not affect the results significantly, and hence the time and spatial steps used were deemed appropriate. The pdepe function in MATLAB used requires all PDEs to have diffusion terms. For those species that are not diffusible, a small diffusivity value (5e-4 μm²/s) was used. This diffusivity parameter was varied, and it was verified that a modest increase did not significantly affect the results (Fig. S4).

Estimation of Arp2/3 complex abundance adjacent to the leading edge

The fluorescence of labeled, endogenous Arpc2 protein adjacent to the leading edge was quantified experimentally. To relate this measurement to the model, we need to account for Arp2/3 complex recruited to the membrane (variable a in the model) and incorporated into the F-actin network. If we refer to the concentration of the latter as A (μM), we calculate its value at the boundary from a flux balance as follows (refer to Text S1 for other definitions; N_Av is Avogadro’s number in matching units).

$$Flux={\left(V_{F}A\right)|}_{x=0}=V_{pol}{A|}_{x=0}=\frac{n_b}{N_{Av}}.$$

At steady state,

$$n_b=k_{cap}\left(b-b_0\right);$$

$${A|}_{x=0}=\frac{k_{cap}}{N_{Av}{V}_{pol}}\left(b-b_0\right).$$

We estimate that the Arpc2 fluorescence one would measure in the leading-edge region (≈0.3 μm) is approximately proportional to the quantity

$$\text{Arp abundance}=a+\left(0.3\text{μm}\right)N_{Av}{A|}_{x=0}=a+\left(0.3\text{μm}\right)\frac{k_{cap}}{V_{pol}}\left(b-b_0\right).$$

This quantity is plotted on the ordinate axis for the simulation results presented in Fig. 4 G.

Figure 4.

The abundance of nascent adhesions and of F-actin near the leading edge are positively correlated and regulated by NMII activity. (A–F) Fibroblasts endogenously expressing Arpc2-mScarlet and stably expressing GFP-paxillin and LifeAct-miRFP670 were plated on glass coated with 100 ng/mL or 10 μg/mL Fn and either untreated (Ctrl) or treated with the NMII inhibitor para-amino-blebbistatin (Bleb; 20 μM). Analyses were performed on n = 16 cells across three separate experiments for each condition. (A) Representative live-cell images; scale bar, 5 μm. (B) Spatial profiles of mean Arpc2-mScarlet intensity from the edge of lamellipodial protrusions. The smaller lines outline the SE of the mean. (C) Quantification of the mean Arpc2-mScarlet intensity along the edge of lamellipodial protrusions. Error bars represent SD. Bleb treatment elicited a significant increase relative to Ctrl in cells plated on 100 ng/mL Fn (p = 0.0004^∗∗∗), and 10 μg/mL Fn (p = 0.0056^∗∗). (D) Quantification of the width of the region that is greater than half of the maximum intensity of Arpc2-mScarlet along a 5-μm-wide line starting from the edge of a lamellipodial protrusion. Error bars represent SD. Bleb treatment reduced the mean value in cells plated on 100 ng/mL Fn (p = 0.0369^∗), but the effect did not pass the significance test for 10 μg/mL Fn (p = 0.351). (E) Quantification of the number of GFP-paxillin punctae per micrometer contour length in the same edge regions measured in (C). Error bars represent SD. Bleb treatment elicited an overall increase in mean nascent-adhesion density, which was significant for cells plated on 100 ng/mL Fn (p = 0.0011^∗∗), but not 10 μg/mL Fn (p = 0.621). (F) Plot of all values collected and shown in (C) against those shown in (E). Together, there is a positive correlation between the enrichment of Arpc2-mScarlet and density of small GFP-paxillin clusters at the edge of lamellipodial protrusions (R = 0.640, p < 0.0001^∗∗∗∗). (G) Steady-state simulation results (same conditions as in Fig. 3) plotting Arp abundance (see Materials and methods) versus area under the curve (AUC) of nascent adhesions density between x = 0 μm and x = 1 μm.

Plasmids

High-fidelity Cas9 (VP12) was a gift from Keith Joung (Addgene plasmid #72247; http://n2t.net/addgene:72247; RRID: Addgene_72247). The guide RNA sequence 5′-GAGGAAGCGCTGTCGACCGA-3′ targeting the genomic region just downstream of the Arpc2 stop codon was expressed from pLKO5.sgRNA.EFS.GFP, a gift from Benjamin Ebert (Addgene plasmid #57822; http://n2t.net/addgene:57822; RRID: Addgene_57822). Homology arms consisting of the 800 bp upstream and downstream of the Arpc2 gRNA target sequence, ordered as gblocks (Integrated DNA Technologies, Coralville, IA), were assembled to flank sequences coding for a GGGGS linker, mScarlet fluorophore, and SspB (nano) in a pBlueScript II backbone using Gibson Assembly. The lentiCRISPR version 2 vector was a gift from Feng Zhang (Addgene plasmid #52961; http://n2t.net/addgene:52961; RRID: Addgene_52961), and the Fn gRNA sequence 5′-GGCTTCGGTGCAGCGCACCG-3′ was inserted for gRNA expression according their protocol. The sequence for LifeAct was encoded in the flanking sequences of primers that were used to move the miRFP670 CDS into a LentiLox5.0 lentiviral expression vector lacking loxP sites via Gibson Assembly. Plasmids for the lentiviral expression of eGFP-Beta-Actin (54) and Pxn-GFP (55) have been previously described.

Cell lines

Mouse embryonic fibroblasts were isolated from mice with Ink4a/Arf−/−, Rosa26 CreERT2, and conditional Arpc2 alleles in the C57BL/6J background (The Jackson Laboratory, Bar Harbor, ME), as previously described and detailed (56). Cell lines were cultured and imaged in Dulbecco’s modified Eagle’s medium (4.5 g/L D-glucose, L-glutamate, sodium pyruvate) supplemented with 10% fetal bovine serum and 1× GlutaMax (Gibco, Waltham, MA). For the labeling of the endogenous Arp2/3 complex, a clonal MEF line that was verified to be diploid (JR20) was transfected using Viromer RED reagent (OriGene, Rockville, MD) with plasmids encoding high-fidelity Cas9 (57), GFP and Arpc2 single guide RNA (both from a pLK05 backbone), and Arpc2-mScarlet-SspB homology repair plasmid. These cells were sorted using fluorescence-activated cell sorting for the expression of GFP to isolate the cells that were successfully transfected. After culturing for 1 week, cells were sorted again for the expression of mScarlet and absence of GFP, adding single cells into individual wells of 96-well plates. A clonal line with biallelic Arpc2-mScarlet was identified and further manipulated to remove the endogenous expression of fibronectin (Fn) via CRISPR knockout of the Fn gene. For this, cells were infected with lentivirus generated using lentiCRISPR version 2 (58) with a guide RNA targeting the mouse Fn gene. Single-cell clones were established using fluorescence-activated cell sorting after puromycin selection, and the loss of Fn expression was verified using genomic sequencing, Western blot, and immunostaining. These MEFs were further manipulated for the stable lentiviral expression of Pxn-GFP and LifeAct-miRFP670 or eGFP-beta-actin. Lentiviral generation and infection were carried out as described previously (59).

Live cell imaging

Before imaging, glass-bottom dishes (MatTek, Ashland, MA) were coated with human Fn (Corning, Corning, NY) at 10 μg/mL or 100 ng/mL diluted in phosphate-buffered saline (PBS) for 1 h at 37°C. Imaging dishes were washed with PBS and seeded with cells in standard culture media 1 h before imaging. Images were captured on a Zeiss LSM800 confocal microscope using a 63× objective. All still images taken to visualize Arpc2-mScarlet, GFP-paxillin, and LifeAct-miRFP670 were captured at the same settings (zoom, averaging, laser power, etc.) to allow the most accurate comparison between treatments. Pharmacological inhibitors of myosin (20 μM para-amino-blebbistatin; Axol Bioscience, Cambridge, UK) and F-actin disassembly (50 nM jasplakinolide; Tocris, Bristol, UK) were added 30 min after seeding cells and remained throughout the duration of imaging. In fluorescence recovery after photobleaching experiments designed for visualization and quantification of actin polymerization and retrograde flow velocities, the leading-edge region was bleached (10 mW, 488 nm laser; 0.67 μs pixel dwell time; 0.76 μm pixel size; 15 iterations); see Video S1 for a representative example. Fluorescence recovery was subsequently monitored at roughly 1.5 s intervals.

Barbed-end assay

The barbed ends of free actin filaments were labeled as previously described but with slight modifications (60). Cells were plated into glass-bottom dishes prepared as described above and allowed to settle for 2 h before labeling and fixation. Cells were washed with prewarmed PBS and then permeabilized and labeled with 3.2 μM Alexa Fluor 488-labeled actin in permeabilization buffer (20 mM HEPES, 138 mM KCl, 4 mM MgCl₂, 3 mM EGTA, 0.2 mg/mL saponin, 1% bovine serum albumin, 1 mM ATP, 3 μM phalloidin) for 30 s. The cells were then immediately fixed using 4% paraformaldehyde and washed in PBS before imaging.

Image quantification and statistical analysis

The Fiji distribution of ImageJ (National Institutes of Health, Bethesda, MD) was used to extract the relevant metrics from live cell imaging data. To measure the average intensity of Arpc2-mScarlet at the edge of protrusions, a region of interest encompassing ∼300 nm just inside and including the cell edge was manually drawn by hand along the edge of lamellipodial protrusions. Distinct punctae of GFP-paxillin roughly 100–300 nm in diameter and within ∼2 μm from the leading edge were manually counted. For profile plots of Arpc2-mScarlet labeling, a 5-μm-long, ∼300-nm-wide line was manually drawn from the cell edge inward into the cell. For each fluorescence recovery after photobleaching measurement, a kymograph was constructed for a rectangle approximately 5 μm long and 1 μm wide, centered near the edge of the protrusion and oriented longwise along the direction of retrograde flow. The distances of polymerization, membrane protrusion, and retrograde flow were thus visualized over time as illustrated in Fig. 5 B, and velocities were calculated as distance/time. Data were organized in Excel (Microsoft, Redmond, WA), and Prism (GraphPad Software, La Jolla, CA) was used for statistical analysis and graphing data. Unpaired, parametric t-tests were used to assess statistical significance, with p values < 0.05 considered to indicate statistical significance.

Figure 5.

Myosin inhibition reduces actin polymerization, cellular protrusion, and retrograde actin flow velocities at the leading edge. Fibroblasts endogenously expressing Arpc2-mScarlet and stably expressing GFP-β-actin were plated on glass coated with either 100 ng/mL or 10 μg/mL Fn and either untreated (Ctrl) or treated with the NMII inhibitor para-amino-blebbistatin (Bleb; 20 μM). (A) Representative live-cell images; scale bar, 5 μm. The top-right insets show kymographs of GFP-labeled actin returning to the region indicated by the red dashed line rectangle in the main image that has been bleached and subsequently monitored. (B) Example kymograph of F-actin returning to the leading edge, annotated to show how velocities of actin polymerization, membrane protrusion, and leading-edge F-actin retrograde flow were estimated. (C–E) Measurements of actin polymerization (C), membrane protrusion (D), and F-actin retrograde flow (E) velocities at the leading edge of cells under different experimental conditions (100 ng/mL Ctrl, n = 31 cells; 100 ng/mL Bleb, n = 57 cells; 10 μg/mL Ctrl, n = 32 cells; 10 μg/mL Bleb, n = 43 cells). Error bars represent SD. (C) Myosin inhibition reduced actin polymerization rates in cells plated on glass coated with 100 ng/mL (p = 0.0348^∗) and 10 μg/mL Fn (p = 0.0147^∗). (D and E) Reductions that did not meet threshold for statistical significance were seen in the velocities of protrusion (100 ng/mL, p = 0.2635; 10 μg/mL, p = 0.1231) and retrograde actin flow (100 ng/mL, p = 0.0676; 10 μg/mL, p = 0.1937).

Results

A spatially extended model integrating adhesion, signaling, and actin dynamics

We developed a mathematical model composed of PDEs and associated boundary conditions, described in detail in Text S1, which treats a cell protrusion as a continuum, with distance from the leading edge, x, as the lone spatial dimension (Fig. 1 A). The model comprises modules describing (1) dendritic actin network dynamics; (2) adhesion dynamics, signaling, and myosin regulation; and (3) integration of F-actin and adhesion dynamics via mechanics. Actin polymerization occurs at the leading edge through conversion of diffusible, ATP-bound G-actin to F-actin on free barbed ends, which are formed by Arp2/3 complex-mediated branching (7) and consumed by capping (29). Arp2/3 complex is activated at the leading edge through Rac signaling, which is mediated by nascent adhesions (clusters of integrins and the proteins in complex with them). In concert, nascent adhesions form in response to membrane protrusion (11), and they are stabilized by transient interactions with F-actin (11,21, 22, 23), a state we refer to as “clutched.” While not engaged with F-actin, nascent adhesions have a high propensity to disintegrate or disassemble. While clutched, nascent adhesions mediate activation of both Rac and Rho GTPases, with the latter promoting activation of NMII (12, 13, 14, 15,61,62). As F-actin flows rearward from the leading edge, it undergoes an aging process (32), and thus it is converted to a form that may be engaged by active NMII and ADF/cofilin, which mediates F-actin severing and depolymerization (53,63, 64, 65). The latter is modeled as a spontaneous conversion of the aged F-actin species to diffusible, ADP-bound G-actin. Cytosolic G-actin is subject to spontaneous ADP/ATP exchange, thus completing the cycle.

Figure 1.

Integrative spatial model of leading-edge cell motility. (A) Model schematic. The model is composed of differential equations; distance from the leading edge, x, is the lone spatial dimension. Nascent adhesions form in response to membrane protrusion and are stabilized by transient interactions with F-actin (clutched). Nascent adhesions mediate activation of both Rac and Rho GTPases, which in turn mediate activation of the Arp2/3 complex and myosin (NMII) through WAVE and ROCK, respectively. Activated Arp2/3 is incorporated into the F-actin network at the leading edge, generating new barbed ends for actin polymerization. F-actin undergoes an aging process, and thus it is converted to a form (F-actin 2) that may be engaged by ADF/cofilin, which mediates F-actin severing and depolymerization and by active myosin II. Mechanical effects include inhibition of actin polymerization by boundary stress, traction force borne by clutched adhesions that resists F-actin retrograde flow, and contractile stress imposed by active myosin II. (B–E) Steady-state spatial profiles of key variables for a base-case simulation with no myosin contractility and leading-edge adhesion density N₀ = 20 μm⁻²: F-actin species (B), G-actin species (C), nascent adhesion and active Rac and Rho species (D), and F-actin velocity relative to the boundary and absolute retrograde flow velocity (E).

The biochemical processes described above are integrated with mechanical effects, namely, inhibition of F-actin polymerization in response to membrane boundary stress, which increases according to the speed of membrane protrusion (40,48,66,67); rearward (retrograde) flow of the F-actin network, which is resisted by the clutched adhesions, which are modeled as springs; dissociation of the F-actin-adhesion linkages, with a propensity that increases exponentially with force (68); and contractile stress imposed by active NMII, according to its spatial gradient (42,43).

As a first pass, parameterization of the model was based on characteristic time, length, and abundance scales (Text S1). Then, as discussed below, we refined key parameters after iterative comparison of the model output and experimental observations. In the base-case set of parameters (Table S1), we set NMII contractility to zero, representing either a NMII-inhibited cell (69) or a region of the cell where NMII is inactive or negatively regulated (70,71); the influence of NMII was built on the foundation of this base case.

In models that have considered adhesion-based signal transduction (45,46,48,52), a principal input is a parameter or variable related to the density of immobile adhesive ligand. In our model, that is the parameter N₀, the density of nascent adhesions at the leading-edge boundary. For base-case parameters and N₀ = 20 μm⁻², steady-state spatial profiles of key variables are shown (Fig. 1 B–E). In this simulation, the membrane densities of barbed ends (82 μm⁻²) and activated Arp2/3 complex (9 μm⁻²) at the leading edge are consistent with evidence (29,72,73), and total F-actin decays in concentration over a distance of ∼3 μm (11,65); the fresh versus aged F-actin species are shown (Fig. 1 B). Depending on the rate of actin polymerization, there is a significant gradient of G-actin (ATP) concentration at the leading edge (29) (Fig. 1 C). Because of the stabilization of nascent adhesions by interacting with F-actin (both forms), the density of nascent adhesions decays over a comparable distance from the leadings edge as does F-actin concentration; Rac- and Rho-GTP generated by adhesion-mediated signal transduction have a somewhat longer spatial reach, on the basis of experimentally estimated diffusion and deactivation parameters for Rac (51) (Fig. 1 D). The flux of F-actin at the leading edge translates to a polymerization velocity ≈0.1 μm/s; roughly half of this velocity manifests as membrane protrusion, the balance as retrograde flow of the F-actin network at the leading edge. These values are consistent with measurements (11,65,69). Because of stresses resisting F-actin flow, the retrograde flow velocity decreases with distance from the leading edge (Fig. 1 E). Note that the retrograde flow velocity is in a fixed frame of reference. The velocity of F-actin relative to the leading edge (x = 0) is the sum of the retrograde flow and membrane protrusion velocities.

Signaling and mechanical feedback between nascent adhesions and F-actin promotes lamellipodial protrusion

To begin exploring the interplay of dynamic processes, we sought to address the mechanosensitive nature of nascent adhesions, which mediate signal transduction and mechanically interact with the F-actin network. To simplify the signaling aspect, we first apply the no-myosin case (as in Fig. 1) and focus on Rac signaling only. To evaluate the roles of adhesions that influence leading-edge motility in this context, we varied the leading-edge adhesion density, N₀ (Fig. 2). The model predicts that the steady-state velocities of membrane protrusion and F-actin retrograde flow at the leading edge are biphasic with respect to N₀ (Fig. 2 A). Moreover, the protrusion velocity exhibits a switch-like increase between N₀ = 14 and N₀ = 16 μm⁻², accompanied by a dramatic increase in the density of F-actin barbed ends (Fig. 2 B). What causes this transition?

Figure 2.

Variation of leading-edge adhesion density with no myosin contractility. (A) Steady-state velocities of membrane protrusion (V_mem), F-actin retrograde flow evaluated at the leading edge (V_ret[x = 0]), and actin polymerization (V_pol) for various values of the leading-edge adhesion density, N₀. These quantities are related by V_mem + V_ret(x = 0) = V_pol. (B) Steady-state values of the barbed-end density at the membrane versus leading-edge adhesion density, N₀. (C–F) Steady-state spatial profiles of key variables for indicated values of the leading-edge adhesion density, N₀: total F-actin (C), total nascent adhesions (D), active Rac (E), and ATP-bound G-actin (F).

At low adhesion densities, adhesions support low concentrations of F-actin (Fig. 2 C), following the barbed-end density trend. The low F-actin density poorly stabilizes the nascent adhesions, and therefore the adhesion density decays more steeply with distance from the leading edge than the F-actin density (Fig. 2 C and D). As a consequence, Rac activation is low (Fig. 2 E), which explains the low barbed-end density for N₀ ≤ 14 μm⁻². This low-adhesion state undermines protrusion in two distinct ways. First, a lower density of barbed ends translates to a higher force per barbed end and thus a lower efficiency of actin polymerization (Fig. 2 B, inset). Second, a lower density of clutched adhesions does not significantly resist retrograde flow. As leading-edge adhesion density is increased above a critical value, positive feedback between adhesions and F-actin drives sensitive increases in barbed-end density, F-actin concentration, adhesion stability, and Rac signaling. The greater efficiency of actin polymerization and the greater resistance to retrograde flow results in increased membrane protrusion (Fig. 2 A). As the leading-edge adhesion density is further increased, the model predicts proportional gains in barbed-end density (Fig. 2 B). However, the velocity of actin polymerization decreases because of progressively depleted G-actin (ATP) (Fig. 2 F), reflecting competition among the increasingly abundant barbed ends. The velocities of membrane protrusion and retrograde flow both decrease in this regime, but the latter decreases more, because the greater abundance of adhesions further resists F-actin flow (Fig. 2 A). This manifests in the F-actin density profiles; with higher adhesion density, but reduced F-actin flow relative to the boundary, F-actin is predicted to be more abundant at the leading edge but with a steeper gradient (Fig. 2 C).

Myosin II contractility mechanically disrupts the feedback between nascent adhesions and F-actin

In the previous analysis, we considered the system with NMII turned off. Active NMII responds to Rho signaling and applies stress on the F-actin network in the direction of increasing actomyosin concentration. As NMII, like ADF/cofilin, can presumably only bind to older F-actin (69), actomyosin abundance increases with distance from the leading edge, and thus myosin contractility promotes retrograde flow. To investigate the influence of this effect, we repeated the variation of leading-edge adhesion density with NMII present (Fig. 3).

Figure 3.

Myosin II contractility mechanically disrupts positive feedback between adhesions and F-actin. (A and B) Steady-state values of actin polymerization velocity (V_pol), membrane protrusion velocity (V_mem) and barbed-end density for various values of the leading-edge adhesion density, N₀, and indicated values of the myosin contractility parameter, α_myo. (C–E) Steady-state spatial profiles of key variables for indicated values of the leading-edge adhesion density, N₀, and high α_myo = 10 pN-μm: total F-actin (C), total nascent adhesions (D), and actomyosin (E). The inset in (D) compares total adhesion profiles for high versus no myosin contractility for N₀ = 18 μm⁻². (F) Steady-state spatial profiles of F-actin retrograde flow velocity at high leading-edge adhesion density (N₀ = 18 μm⁻²), for zero and high α_myo.

At low leading-edge adhesion densities, the presence of NMII does not substantially affect membrane protrusion or leading-edge retrograde flow (Fig. 3 A). At higher leading-edge adhesion densities, NMII contractility has a strong influence, curtailing the sensitive increases in adhesion-mediated signaling and barbed-end density (Fig. 3 A and B); with barbed-end density reduced, so is competition for G-actin, and actin polymerization velocity is predicted to be higher with NMII present (Fig. 3 A). The net effect is reduced F-actin concentration (Fig. 3 C). Without the feedback between adhesions and F-actin engaged, the stabilization of adhesions does not occur, and nascent adhesion density decays precipitously with distance from the leading edge, even at optimal N₀ (Fig. 3 D), attributed to increased engagement of actomyosin as a negative feedback (Fig. 3 E). With less resistance to F-actin flow from clutched adhesions, the retrograde flow velocity at the leading edge is higher in the presence of NMII contractility (Fig. 3 F). Thus, the model predicts that contractile stress exerted by NMII on the F-actin network mechanically disrupts the positive feedback between nascent adhesions and F-actin, by destabilizing the former.

To further explore the mechanical stability of the nascent adhesion/F-actin clutch, we varied the mechanical stiffness of those bonds, characterized by a spring constant, κ_c, as in other modeling studies (37, 38, 39,74,75). The higher its value, the higher the force on the bond for a given displacement (or, equivalently, for a given duration under constant retrograde flow). The results show that increasing adhesive bond stiffness can either enhance or diminish protrusion, depending on whether the system is in a regime where the clutch can bear more mechanical load or a regime where more mechanical load causes the bonds to rupture sooner (Fig. S1).

Experimental measurements show a correlation between nascent adhesion density and leading-edge recruitment of Arp2/3 complex, consistent with model predictions

Modeling results suggest a mutually positive relationship between nascent adhesion formation/stability and the generation of F-actin barbed ends mediated by the Arp2/3 complex, with ECM density as a key input and NMII activity as key regulator. To test these basic predictions, we performed live imaging of the endogenous Arp2/3 complex in mouse embryonic fibroblasts in the presence or absence of a NMII inhibitor (Fig. 4). These cells additionally express markers for F-actin (LifeAct) and adhesions (paxillin) and carry mutations that prevent the secretion of their own fibronectin (Fn), facilitating careful analysis of protrusion machinery on different ECM coating conditions (100 ng/mL and 10 μg/mL Fn; Fig. 4 A). In a cohort of 16 cells per condition (three independent experiments), the spatial profile of Arpc2 fluorescence at the leading edge was quantified. Arpc2 fluorescence reflects its recruitment at the leading edge and its incorporation in the dendritic F-actin network, and we confirmed that peak Arpc2 fluorescence correlates well with labeling of free barbed ends, measured in the same regions (Fig. S2). The results show that the peak Arpc2 fluorescence increases, and the profile tends to narrow, as Fn density is increased or if NMII is inhibited (Fig. 4 B–D). These trends are in qualitative agreement with the predicted F-actin profiles shown in Figs. 2 C and 3 C. The measured increase in Arpc2 intensity and density is accompanied by higher nascent adhesion density within the same region, ∼300 nm of the protrusive edge (Fig. 4 E). For all of cells across the four experimental conditions, a plot of peak Arpc2 fluorescence versus nascent adhesion density shows a positive correlation (R = 0.640, p < 0.0001^∗∗∗∗) (Fig. 4 F). This near-proportional relationship is captured by the model for the conditions shown in Fig. 3, using an estimation of the abundance of Arp2/3 molecules in the edge region (see Materials and methods; Fig. 4 G). It should be mentioned, however, that the model predicts a far greater sensitivity to NMII inhibition at higher ECM density; in contrast, the experimental measurements suggest that nascent adhesion abundance plateaus under favorable conditions.

Experimentally measured F-actin polymerization, membrane protrusion, and F-actin retrograde flow velocities follow predicted trends for increasing ECM density but not for myosin II inhibition

Whereas nascent adhesions, barbed ends, and F-actin tend to be positively correlated, gains in these “static” readouts are not necessarily accompanied by higher velocities of F-actin polymerization and membrane protrusion. Indeed, as the abundance of nascent adhesions is increased, well above the critical value required for substantial protrusion, the model predicts that actin polymerization velocity decreases, while membrane protrusion velocity shows little change or a modest decrease. To address this experimentally, we acquired time-lapse images showing the dynamic reincorporation and flow of fluorescent β-actin after photobleaching the leading edge, for the same set of environmental conditions applied prior (Fig. 5; see also Video S1). With this approach, the velocities of F-actin polymerization, membrane protrusion, and F-actin retrograde flow are quantified (Fig. 5 A and B). In qualitative agreement with the model, increasing Fn density yields lower polymerization and retrograde flow velocities, while membrane protrusion shows no apparent change, on average (Fig. 5 C–E). However, examining the effects of NMII inhibition was not as well supportive of the model. Although the treatment yields significantly lower polymerization velocity as expected (Fig. 5 C), the predicted increase in membrane protrusion is absent (Fig. 5 D), as the effect on reducing retrograde flow was far less prominent than predicted (and lacking statistical significance; Fig. 5 E). As we discuss later, such a discrepancy can offer insights into limitations of the model as well as the interpretation of experiments.

G-actin fuels the feedback between adhesions and F-actin

In concert with the mechanical effects considered above, an important facet of the model is its accounting of F-actin flux and turnover. To modulate the stabilizing effect of F-actin on nascent adhesions, we repeated the variation of leading-edge adhesion density with higher and lower concentrations of total intracellular actin, in the no-myosin limit (Fig. 6). At low adhesion densities (N₀ ≤ 10 μm⁻²), actin polymerization velocity proportionally increases with total actin abundance (Fig. 6 A); however, most of the gain in actin polymerization manifests as retrograde flow, and therefore membrane protrusion is low in this regime (Fig. 6 B). As leading-edge adhesion density is increased, and the positive feedback between nascent adhesions and F-actin is engaged, the abundance of actin substantially influences membrane protrusion. Both the maximum protrusion velocity and the critical value of N₀ where positive feedback is engaged are sensitive to the availability of G-actin (Fig. 6 B). The trend is similar for barbed-end density, except that this quantity is not as sensitive to a doubling of the actin abundance, relative to the base case (Fig. 6 C). This is because the stability of the nascent adhesions, and thus the magnitude of Rac signaling, is only modestly improved at high adhesion density.

Figure 6.

G-actin fuels positive feedback between adhesions and F-actin. Steady-state values of actin polymerization velocity (A), membrane protrusion velocity (B), and barbed-end density (C) as a function of leading-edge adhesion density, N₀, for different values of the initial G-actin concentration, G_T (asterisk indicates the base-case value), and with no NMII activity (α_myo = 0).

F-actin turnover enhances positive feedback via G-actin flux, not by reducing myosin stress

Although it is intuitive that availability of G-actin enhances motility, the effect of modulating turnover of aged F-actin was not. Older F-actin is able to recruit NMII, and both new and aged F-actin are assumed to stabilize nascent adhesions; therefore, it was not obvious whether turnover of aged F-actin would enhance or reduce leading-edge protrusion. This question is also motivated by the prominent function of cofilin in cell migration (76, 77, 78). Whereas overexpression or activation of cofilin would amplify F-actin turnover, treatment with the compound jasplakinolide (Jasp), which binds F-actin and prevents cofilin severing, inhibits it. To assess the effect of F-actin turnover in the model, we varied the corresponding rate constant, k_F2, by a factor of 10 in each direction, with NMII absent or present (Figs. 7 and 8). In the absence of myosin contractility, increasing F-actin turnover results in subtle changes in membrane protrusion, switching from a modest decrease to a modest increase as the ECM parameter N₀ is increased; in contrast, inhibition of turnover substantially diminishes protrusion (Fig. 7 A). These trends are also reflected in the velocity of actin polymerization (Fig. 7 B). At high adhesion density, the intermediate, base-case value of k_F2 = 0.1 s⁻¹ yields the greatest adhesion stability (Fig. 7 C) and Rac signaling that determines barbed-end density (Fig. 7 D), but those advantages are countered by suboptimal concentration of G-actin (ATP) at the leading edge (Fig. 7 E). Like the velocities of protrusion and polymerization at high adhesion density, retrograde flow velocity at the leading edge increases as F-actin turnover is increased; retrograde flow is dissipated most efficiently for the base-case value, because of the greater stability of adhesions (Fig. 7 F).

Figure 7.

In the absence of NMII activity, increasing F-actin turnover strengthens adhesion/F-actin feedback by increasing G-actin flux. The F-actin-2 turnover rate constant, k_F2, was varied as indicated (asterisk indicates the base-case value) in the absence of NMII activity (α_myo = 0). (A and B) Steady-state velocities of membrane protrusion (A) and actin polymerization (B) as a function of leading-edge adhesion density, N₀. (C–F) Steady-state spatial profiles of total nascent adhesion density (C), active Rac (D), ATP-bound G-actin (E), and F-actin retrograde flow velocity (F) for leading-edge adhesion density N₀ = 18 μm⁻².

Figure 8.

With NMII activation, increasing F-actin turnover enhances protrusion via adhesion/F-actin feedback but with complex effects on NMII activity and actin retrograde flow. The F-actin-2 turnover rate constant, k_F2, was varied as indicated (asterisk indicates the base-case value) in the presence of intermediate NMII activity (α_myo = 1 pN-μm). (A and B) Steady-state velocities of membrane protrusion (A) and actin polymerization (B) as a function of adhesion density, N₀. (C–H). Steady-state spatial profiles of total nascent adhesion density (C), active Rac (D), ATP-bound G-actin (E), aged F-actin (F), actomyosin (G), and F-actin retrograde flow velocity (H) for leading-edge adhesion density N₀ = 18 μm⁻².

In the presence of myosin contractility (Fig. 8), increased/decreased F-actin turnover generally enhances/diminishes membrane protrusion, with greater sensitivity than the no-myosin case (Fig. 8 A, compare with Fig. 7 A). This trend is largely reflected in the effect on F-actin polymerization (Fig. 8 B). The positive influence of F-actin turnover on protrusion velocity is reflected by gains in adhesion stability (Fig. 8 C), Rac signaling (Fig. 8 D), and G-actin flux (Fig. 8 E), the hallmarks of positive feedback between adhesions and F-actin. With enhanced F-actin turnover, the abundance of aged F-actin and the ability to recruit active NMII are reduced; does this contribute to the stronger positive feedback? For high adhesion density and the set of constant parameter values considered, not necessarily. Although increasing F-actin turnover negatively affects the abundance of aged F-actinm as expected (Fig. 8 F), the increase in adhesion stability results in enhanced Rho signaling (similar to the enhancement of Rac signaling shown in Fig. 8 D). Consequently, there is an optimal condition for recruitment and activation of NMII, with the base-case value showing the greatest magnitude (Fig. 8 G). However, evaluating the gradient of NMII activity, which determines the stress on the network, the gradient close to the leading edge is actually steepest for the enhanced turnover case (Fig. 8 G). The net mechanical effect of clutched adhesions and NMII contractility manifests in the profile of F-actin retrograde flow, with complex results (Fig. 8 H). For example, at the membrane edge, retrograde flow velocity for reduced F-actin turnover is only slightly lower than the base case. This outcome is the net effect of reduced actin polymerization, reduced clutch resistance, and reduced NMII contractility.

Treatment with the F-actin stabilizer Jasp reduces F-actin protrusion and polymerization, consistent with model predictions

To test the effect of altered F-actin turnover rate, predicted by the model, we performed experiments to measure changes in actin dynamics in cells treated with the aforementioned F-actin stabilizer, Jasp (Fig. 9). Using a moderate concentration of Jasp (50 nM) and measurements of fluorescent β-actin flow after photobleaching (Fig. 9 A), we found that Jasp treatment significantly reduced the mean F-actin polymerization and protrusion velocities (Fig. 9 B and C), consistent with the model predictions. Jasp also significantly reduced the mean retrograde flow velocity (Fig. 9 D), which is also consistent with the model predictions, especially when NMII was assumed to be inactive. To probe the measurements in more detail, we examined the correlation of polymerization and protrusion velocities for the control and Jasp groups (Fig. 9 E). The Jasp group shows a much tighter correlation; the difference between polymerization and protrusion velocities is the retrograde flow (evaluated at the leading edge), which is the horizontal distance from the identity line (Fig. 9 E, black dashed line). The tighter correlation of the Jasp measurements, with slope closer to 1, reflects a more consistent retrograde flow. Although the majority of the control measurements fall close to the same trend, the control measurements include more outliers, which skew the correlation (Fig. 9 E) and the comparison of retrograde flow velocities (Fig. 9 D). A model-guided hypothesis is that the control measurements reflect greater variability on the cusp of activating positive feedback and/or greater sensitivity to NMII contractility, which are quelled when F-actin turnover/flux is inhibited.

Figure 9.

Stabilization of F-actin reduces actin polymerization and membrane protrusion velocities at cell protrusions. Fibroblasts endogenously expressing Arpc2-mScarlet and stably expressing GFP-β-actin were plated on glass coated with 10 μg/mL Fn and either untreated (Control) or treated with the F-actin stabilizer Jasp (50 nM). Fluorescence recovery after photobleaching (FRAP) analyses were performed (n = 51 cells for control and n = 42 cells for Jasp, across three separate experiments for each condition). (A) Representative live-cell images; scale bar, 5 μm. The top-right insets show kymographs of GFP-labeled actin returning to the region indicated by the red dashed line rectangle in the main image that has been bleached and subsequently monitored. (B–D) Measurements of actin polymerization (B), membrane protrusion (C), and F-actin retrograde flow (D) velocities show significant reduction of the mean value with Jasp treatment. Error bars represent SD, and p values are <0.0001^∗∗∗, 0.0347^∗, and 0.0002^∗∗∗, respectively. (E) Plot of all values collected and shown in (C) against those shown in (B) for the control (R = 0.5105, p = 0.0001^∗∗∗) and Jasp (R = 0.7991, p < 0.0001^∗∗∗) groups. The identity (y = x) line is indicated; a point’s horizontal distance from the identity line is the velocity of leading-edge F-actin retrograde flow.

Discussion

In this study, we have constructed an integrative, PDE model that includes both biochemical regulation and mechanics of the dendritic F-actin network, with nascent adhesions as the key interface for both aspects. The model predicts that membrane protrusion requires a set of optimal conditions, including ECM density, myosin contractility, adhesion bond stiffness, and F-actin turnover for maximal protrusion velocity, consistent with the literature (45,52,79,80). At lower ECM densities, the abundance of nascent adhesions is too low to support effective signal transduction and to resist retrograde flow. With increasing ECM density, substantial gains in membrane protrusion rely on a positive feedback loop, in which adhesion-mediated activation of Rac, a proxy for signaling pathways that activate the Arp2/3 complex, fosters increased formation of barbed ends and F-actin, enhancing membrane protrusion, formation of new adhesions, and stabilization of existing ones. Myosin contractility, buffered by clutched adhesions, opposes membrane protrusion at high ECM density, yielding increased F-actin retrograde flow. By applying stress on the F-actin network, NMII can mechanically destabilize nascent adhesions, weakening the positive feedback between nascent adhesions and F-actin. Increased F-actin turnover, mediated by ADF/cofilin, offsets the influence of NMII.

On the influence of myosin contractility

Multiple experimental observations implicate NMII activity in the mechanical regulation of membrane protrusion. Treatment of fibroblasts with an inhibitor of myosin light chain kinase caused an increase in lamellipodial protrusion (81). In NMII-deficient CHO-K1 cells, protrusion rates were elevated more than 2-fold relative to control, and lamellipodia were broader (69). In fibroblasts migrating on a gradient of fibronectin, inhibition of ROCK increased both migration speed and persistence (82). Although it may seem sensible that pulling on the F-actin network counters the pushing force associated with membrane protrusion, our model suggests that this is a simplistic view. Rather, our simulations predict that myosin contractility adds to the stress caused by F-actin polymerization against the membrane boundary, enhancing retrograde flow that mechanically weakens adhesion/F-actin linkages. This also matches experimental results in which inhibition of NMII activity led to a decrease in retrograde velocity in neural growth cones (83) and fibroblasts (17). Our model predictions are in line with the observations cited above; however, in our cells, NMII inhibition did not enhance protrusion, and retrograde flow was reduced but not significantly (Fig. 5), pointing to certain limitations of the model discussed below.

Conceptually, the view that myosin contractility weakens adhesion/F-actin linkages is similar to those represented in adhesion clutch models (37, 38, 39), but the present model differs in that it connects adhesion stability to signal transduction, which affects the nucleation of barbed ends via the Arp2/3 complex and thus the ability of the F-actin network to collectively overcome the boundary stress. A key prediction in that regard is that NMII contractility indirectly disrupts Rac signaling (and other integrin-mediated signaling), whereas NMII inhibition would be permissive for integrin-mediated signaling. Our live-cell imaging measurements (Fig. 4) support this view. Vicente-Manzanares et al. (84) also showed that after initial Rac signaling by nascent adhesions, nascent adhesions interact with NMIIA and NMIIB, mature and move to the rear of the cell, or more accurately the cell “moves over” the adhesions. Notably, once the adhesions interact with myosin, Rac signaling is muted. Although our model focuses on leading-edge protrusion, NMII also has established roles in adhesion maturation (11,69,85) and F-actin turnover (86). Clearly, NMII is a critical player that influences overall cell migration in multiple ways, which can complicate the interpretation of experiments perturbing its activation or activity. For example, if NMII inhibition reduces F-actin turnover, that effect would counter the gain in adhesion stability, which might explain our experimental findings.

On the influence of adhesive bond stiffness

Appreciation of matrix/substratum stiffness as a key variable affecting mechanosensing and cell migration has grown over the past two decades (87, 88, 89). Stiffer matrices foster higher cellular traction forces and are associated with greater metastasis potential (90,91). Recently, it was demonstrated that migration of cancer cells is biased by a gradient of substratum stiffness and that lamellipodial Arp2/3 is required for this process (92). In the present model, the rigidity of the nascent adhesion-F-actin linkages reflects the stiffness of the substratum and the mechanical compliance of the various molecular interactions associated with the adhesion clutch. This parameter influences how much energy the clutched adhesions dissipate under load associated with F-actin retrograde flow and the overall stability of nascent adhesions. A stiffer linkage is better until its stability is compromised, and so there is an optimum. The relative concentration and spatial profile of F-actin and the spatial profile of nascent adhesions are consistent with those reported for spreading fibroblasts on soft and stiff substrates (93); however, the interpretation presented in that study, based on other experiments, is that the productivity of adhesion formation is the primary determinant. This process is reflected in the parameter N₀ of our model, and its dependence on substrate stiffness suggests a catch-bond (also called a catch-slip bond) character of the adhesion assembly (94). Other recent evidence, interpreted through the use of theoretical modeling, reinforced that the optimal substrate stiffness for cell motility can be shifted by modifying NMII activity and/or adhesion density, consistent with our findings (95). Taken together, our simulations and the experimental evidence outlined above clearly implicate the lamellipodium as a substrate stiffness sensor.

On the influence of F-actin turnover

The abundance of F-actin is important for stabilizing nascent adhesions (11) and is affected by the density of barbed ends, the velocity of actin polymerization, and F-actin severing/turnover (28). Our simulations show that F-actin turnover tends to enhance protrusion by enhancing (or mitigating the reduction of) G-actin flux. Although this aspect has been explored through analysis of other models (29,34,96), the insight gained from the present model is that any such positive influence on protrusion may be compounded through adhesion-mediated Rac signaling. Thus, under conditions in which myosin contractility hinders protrusion, increasing F-actin turnover offsets this effect, but not by reducing myosin stress; in fact, the simulations predict higher contractility when F-actin turnover is increased. Nascent adhesions mediate both Rac and Rho signaling, and the latter compensates for the reduction in aged F-actin.

Cofilin severs F-actin filaments in a concentration-dependent manner, with each scission creating an uncapped barbed end and an uncapped pointed end (53). Cofilin has been implicated in enhanced cell migration (32,76) and cancer invasiveness and metastasis (76, 77, 78,97). Consistent with our model predictions, increased/decreased cofilin activity has been associated with increases/decreases in the abundance of G-actin in the cytosol (98,99) and the rate of actin polymerization (100). In both conceptual and mathematical models of cofilin’s influence on F-actin dynamics (32,53,63,101), a major contribution of actin severing to actin polymerization is the direct generation of barbed ends. Our model neglects this contribution and instead attributes gains in barbed-end density at the leading edge to increased Rac signaling. Contrarily, the aforementioned model by Tania et al. (32) decouples cofilin-mediated F-actin severing and the overall turnover of F-actin, and it considers the G-actin concentration fixed. The compromises made in the two models permit their closure, but a more comprehensive model would include all relevant aspects, perhaps with resolution at the level of individual actin filaments (34). A key question is how effective cofilin-generated barbed ends are at reaching the leading edge; although they are expected to polymerize faster than barbed ends at the boundary, they are also capped at a higher frequency (29,73) and subject to twinfilin-mediated depolymerization (102). Other aspects that could be explored with regard to F-actin turnover are the competition between cofilin and NMII for aged F-actin (103) and the aforementioned disintegration of F-actin by NMII-induced buckling forces (104,105).

Extending the model

The model presented here was formulated with both integration and economy in mind. Incorporating adhesion, signaling, and cytoskeletal dynamics, along with mechanics, demands a level of restraint to avoid parameter bloat. In addition to the possible refinements already mentioned, more details related to integrin-mediated signaling, considering the specifics of Rac/WAVE and Rho/ROCK activation or/and the influence of other signaling pathways, could be added. At the level of adhesion, a more comprehensive model could include details related to integrin clustering and the molecular components responsible for clutching F-actin (106). In that regard, we consider it paramount to parse the knowledge of the field regarding nascent versus mature adhesions, the architecture of the latter being much better understood (94,107,108). Until a unifying theory of adhesion assembly and maturation is approached, it is important to distinguish the classes of adhesions and avoid ascribing knowledge about one to another. At the level of cytoskeletal dynamics, a more comprehensive model could include finer details of actin nucleation via incorporation of the Arp2/3 complex; perhaps more critically, the present model does not explicitly include actin polymerase activities, which are subject to biochemical and mechanical regulation (109, 110, 111), instead treating F-actin elongation as a mass-action process. Other directions to potentially build out from the present model include moving beyond the level of a single lamellipodium and/or allowing for stochastic transitions. Modeling whole-cell migration is clearly challenging, and current successful models have been predicated on a constant cell shape or/and greatly simplified dynamics (112, 113, 114, 115, 116, 117, 118). Similarly, stochastic models have required certain simplifications (40,48,66). Although models such as ours suggest ways to integrate multiple aspects of cell motility, especially where integrin-mediated adhesion and signaling are fundamental, the exercise also illuminates substantial gaps in our understanding.

Author contributions

A.C. designed research, performed research, analyzed data, and wrote and edited the paper. M.T.B. designed research, performed research, analyzed data, and edited the paper. J.E.B. conceptualized the study, designed research, supervised the research, acquired funding, and edited the paper. J.M.H. conceptualized the study, designed research, analyzed data, supervised the research, acquired funding, and wrote and edited the paper.

Editor: Dimitrios Vavylonis.