Physical model for self-organization of actin cytoskeleton and adhesion complexes at the cell front

Directed cell invasion and asymmetric adhesion drive tissue elongation and turning in C. elegans gonad morphogenesis

Development of the C. elegans gonad has long been studied as a model of organogenesis driven by collective cell migration. A somatic cell named the distal tip cell (DTC) is thought to serve as the leader of following germ cells; yet, the mechanism for DTC propulsion and maneuvering remains elusive. Here, we demonstrate that the DTC is not self-propelled but rather is pushed by the proliferating germ cells. Proliferative pressure pushes the DTC forward, against the resistance of the basement membrane in front. The DTC locally secretes metalloproteases that degrade the impeding membrane, resulting in gonad elongation. Turning of the gonad is achieved by polarized DTC-matrix adhesions. The asymmetrical traction results in a bending moment on the DTC. Src and Cdc42 regulate integrin adhesion polarity, whereas an external netrin signal determines DTC orientation. Our findings challenge the current view of DTC migration and offer a distinct framework to understand organogenesis.

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

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.

Methods for assessment of membrane protrusion dynamics

Membrane protrusions are a critical facet of cell function. Mediating fundamental processes such as cell migration, cell-cell interactions, phagocytosis, as well as assessment and remodeling of the cell environment. Different protrusion types and morphologies can promote different cellular functions and occur downstream of distinct signaling pathways. As such, techniques to quantify and understand the inner workings of protrusion dynamics are critical for a comprehensive understanding of cell biology. In this chapter, we describe approaches to analyze cellular protrusions and correlate physical changes in cell morphology with biochemical signaling processes. We address methods to quantify and characterize protrusion types and velocity, mathematical approaches to predictive models of cytoskeletal changes, and implementation of protein engineering and biosensor design to dissect cell signaling driving protrusive activity. Combining these approaches allows cell biologists to develop a comprehensive understanding of the dynamics of membrane protrusions.

Discrete mechanical model of lamellipodial actin network implements molecular clutch mechanism and generates arcs and microspikes

Mechanical forces, actin filament turnover, and adhesion to the extracellular environment regulate lamellipodial protrusions. Computational and mathematical models at the continuum level have been used to investigate the molecular clutch mechanism, calculating the stress profile through the lamellipodium and around focal adhesions. However, the forces and deformations of individual actin filaments have not been considered while interactions between actin networks and actin bundles is not easily accounted with such methods. We develop a filament-level model of a lamellipodial actin network undergoing retrograde flow using 3D Brownian dynamics. Retrograde flow is promoted in simulations by pushing forces from the leading edge (due to actin polymerization), pulling forces (due to molecular motors), and opposed by viscous drag in cytoplasm and focal adhesions. Simulated networks have densities similar to measurements in prior electron micrographs. Connectivity between individual actin segments is maintained by permanent and dynamic crosslinkers. Remodeling of the network occurs via the addition of single actin filaments near the leading edge and via filament bond severing. We investigated how several parameters affect the stress distribution, network deformation and retrograde flow speed. The model captures the decrease in retrograde flow upon increase of focal adhesion strength. The stress profile changes from compression to extension across the leading edge, with regions of filament bending around focal adhesions. The model reproduces the observed reduction in retrograde flow speed upon exposure to cytochalasin D, which halts actin polymerization. Changes in crosslinker concentration and dynamics, as well as in the orientation pattern of newly added filaments demonstrate the model's ability to generate bundles of filaments perpendicular (actin arcs) or parallel (microspikes) to the protruding direction.

Vimentin intermediate filaments modulate cell traction force but not cell sensitivity to substrate stiffness

The ability of cells to sense and respond to the mechanical stiffness of the surrounding matrix is important to support normal cell function, wound healing, and development. Central to the process of durosensing is the cytoskeleton composed of three classes of filaments: F-actin, microtubules, and intermediate filaments (IFs). Vimentin is an IF protein that contributes significantly to cell mechanics and cell traction force, which is required to probe extracellular matrix. The role of vimentin in how cells sense and respond to the mechanical rigidity of extracellular matrix is largely unclear. To investigate the role of vimentin in durosensing, we knocked down the vimentin expression level in 3T3 fibroblasts using shRNA transfection and measured cellular responses as functions of substrate stiffness. We quantified durosensitivity by the rates at which cell area and traction force change with substrate stiffness. Our results show that that vimentin plays a role in durosensing by modulating traction force and knocking out vimentin did not significantly affect durosensitivity. These results indicate that vimentin may be a redundant component of the machinery that cells use to sense substrate stiffness.

Stick-slip model for actin-driven cell protrusions, cell polarization, and crawling

Cell crawling requires the generation of intracellular forces by the cytoskeleton and their transmission to an extracellular substrate through specific adhesion molecules. Crawling cells show many features of excitable systems, such as spontaneous symmetry breaking and crawling in the absence of external cues, and periodic and propagating waves of activity. Mechanical instabilities in the active cytoskeleton network and feedback loops in the biochemical network of activators and repressors of cytoskeleton dynamics have been invoked to explain these dynamical features. Here, I show that the interplay between the dynamics of cell-substrate adhesion and linear cellular mechanics is sufficient to reproduce many nonlinear dynamical patterns observed in spreading and crawling cells. Using an analytical formalism of the molecular clutch model of cell adhesion, regulated by local mechanical forces, I show that cellular traction forces exhibit stick-slip dynamics resulting in periodic waves of protrusion/retraction and propagating waves along the cell edge. This can explain spontaneous symmetry breaking and polarization of spreading cells, leading to steady crawling or bipedal motion, and bistability, where persistent cell motion requires a sufficiently strong transient external stimulus. The model also highlights the role of membrane tension in providing the long-range mechanical communication across the cell required for symmetry breaking.

Cell Shape and Durotaxis Explained from Cell-Extracellular Matrix Forces and Focal Adhesion Dynamics

Many cells are small and rounded on soft extracellular matrices (ECM), elongated on stiffer ECMs, and flattened on hard ECMs. Cells also migrate up stiffness gradients (durotaxis). Using a hybrid cellular Potts and finite-element model extended with ODE-based models of focal adhesion (FA) turnover, we show that the full range of cell shape and durotaxis can be explained in unison from dynamics of FAs, in contrast to previous mathematical models. In our 2D cell-shape model, FAs grow due to cell traction forces. Forces develop faster on stiff ECMs, causing FAs to stabilize and, consequently, cells to spread on stiff ECMs. If ECM stress further stabilizes FAs, cells elongate on substrates of intermediate stiffness. We show that durotaxis follows from the same set of assumptions. Our model contributes to the understanding of the basic responses of cells to ECM stiffness, paving the way for future modeling of more complex cell-ECM interactions.

Extracellular Matrix Geometry and Initial Adhesive Position Determine Stress Fiber Network Organization during Cell Spreading

Three-dimensional matrices often contain highly structured adhesive tracks that require cells to turn corners and bridge non-adhesive areas. Here, we investigate these complex processes using micropatterned cell adhesive frames. Spreading kinetics on these matrices depend strongly on initial adhesive position and are predicted by a cellular Potts model (CPM), which reflects a balance between adhesion and intracellular tension. As cells spread, new stress fibers (SFs) assemble periodically and parallel to the leading edge, with spatial intervals of ∼2.5 μm, temporal intervals of ∼15 min, and characteristic lifetimes of ∼50 min. By incorporating these rules into the CPM, we can successfully predict SF network architecture. Moreover, we observe broadly similar behavior when we culture cells on arrays of discrete collagen fibers. Our findings show that ECM geometry and initial cell position strongly determine cell spreading and that cells encode a memory of their spreading history through SF network organization.

Stick-slip dynamics of migrating cells on viscoelastic substrates

Stick-slip motion, a common phenomenon observed during crawling of cells, is found to be strongly sensitive to the substrate stiffness. Stick-slip behaviors have previously been investigated typically using purely elastic substrates. For a more realistic understanding of this phenomenon, we propose a theoretical model to study the dynamics on a viscoelastic substrate. Our model, based on a reaction-diffusion framework, incorporates known important interactions such as retrograde flow of actin, myosin contractility, force-dependent assembly, and disassembly of focal adhesions coupled with cell-substrate interaction. We show that consideration of a viscoelastic substrate not only captures the usually observed stick-slip jumps but also predicts the existence of an optimal substrate viscosity corresponding to maximum traction force and minimum retrograde flow which was hitherto unexplored. Moreover, our theory predicts the time evolution of individual bond force that characterizes the stick-slip patterns on soft versus stiff substrates. Our analysis also elucidates how the duration of the stick-slip cycles are affected by various cellular parameters.

Modeling Cell Migration Mechanics

Cell migration is the physical movement of cells and is responsible for the extensive cellular invasion and metastasis that occur in high-grade tumors. Motivated by decades of direct observation of cell migration via light microscopy, theoretical models have emerged to capture various aspects of the fundamental physical phenomena underlying cell migration. Yet, the motility mechanisms actually used by tumor cells during invasion are still poorly understood, as is the role of cellular interactions with the extracellular environment. In this chapter, we review key physical principles of cytoskeletal self-assembly and force generation, membrane tension, biological adhesion, hydrostatic and osmotic pressures, and their integration in mathematical models of cell migration. With the goal of modeling-driven cancer therapy, we provide examples to guide oncologists and physical scientists in developing next-generation models to predict disease progression and treatment.

Forces and constraints controlling podosome assembly and disassembly

Podosomes are a singular category of integrin-mediated adhesions important in the processes of cell migration, matrix degradation and cancer cell invasion. Despite a wealth of biochemical studies, the effects of mechanical forces on podosome integrity and dynamics are poorly understood. Here, we show that podosomes are highly sensitive to two groups of physical factors. First, we describe the process of podosome disassembly induced by activation of myosin-IIA filament assembly. Next, we find that podosome integrity and dynamics depends upon membrane tension and can be experimentally perturbed by osmotic swelling and deoxycholate treatment. We have also found that podosomes can be disrupted in a reversible manner by single or cyclic radial stretching of the substratum. We show that disruption of podosomes induced by osmotic swelling is independent of myosin-II filaments. The inhibition of the membrane sculpting protein, dynamin-II, but not clathrin, resulted in activation of myosin-IIA filament formation and disruption of podosomes. The effect of dynamin-II inhibition on podosomes was, however, independent of myosin-II filaments. Moreover, formation of organized arrays of podosomes in response to microtopographic cues (the ridges with triangular profile) was not accompanied by reorganization of myosin-II filaments. Thus, mechanical elements such as myosin-II filaments and factors affecting membrane tension/sculpting independently modulate podosome formation and dynamics, underlying a versatile response of these adhesion structures to intracellular and extracellular cues. This article is part of a discussion meeting issue 'Forces in cancer: interdisciplinary approaches in tumour mechanobiology'.

Predicting Collective Migration of Cell Populations Defined by Varying Repolarization Dynamics

Collective migration of heterogeneous cell populations is an essential aspect of fundamental biological processes, including morphogenesis, wound healing, and tumor invasion. Through experiments and modeling, it has been shown that cells attain front-rear polarity, generate forces, and form adhesions to migrate. However, it remains unclear how the ability of individual cells in a population to dynamically repolarize themselves into new directions could regulate the collective response. We present a vertex-based model in which each deformable cell randomly chooses a new polarization direction after every defined time interval, elongates, proportionally generates forces, and causes collective migration. Our simulations predict that cell types that repolarize at longer time intervals attain more elongated shapes, migrate faster, deform the cell sheet, and roughen the leading edge. By imaging collectively migrating epithelial cell monolayers at high temporal resolution, we found longer repolarization intervals and elongated shapes of cells at the leading edge compared to those within the monolayer. Based on these experimental measurements and simulations, we defined aggressive mutant leader cells by long repolarization interval and minimal intercellular contact. The cells with frequent and random repolarization were defined as normal cells. In simulations with uniformly dispersed leader cells in a normal cell population at a 1:10 ratio, the resulting migration and deformation of the heterogeneous cell sheet remained low. However, when the 10% mutant leaders were placed only at the leading edge, we predicted a rise in the migration of an otherwise normal cell sheet. Our model predicts that a repolarization-based definition of leader cells and their placement within a healthy population can generate myriad modes of collective cell migration, which can enhance our understanding of collective cell migration in disease and development.

Steps in Mechanotransduction Pathways that Control Cell Morphology

It is increasingly clear that mechanotransduction pathways play important roles in regulating fundamental cellular functions. Of the basic mechanical functions, the determination of cellular morphology is critical. Cells typically use many mechanosensitive steps and different cell states to achieve a polarized shape through repeated testing of the microenvironment. Indeed, morphology is determined by the microenvironment through periodic activation of motility, mechanotesting, and mechanoresponse functions by hormones, internal clocks, and receptor tyrosine kinases. Patterned substrates and controlled environments with defined rigidities limit the range of cell behavior and influence cell state decisions and are thus very useful for studying these steps. The recently defined rigidity sensing process provides a good example of how cells repeatedly test their microenvironment and is also linked to cancer. In general, aberrant extracellular matrix mechanosensing is associated with numerous conditions, including cardiovascular disease, aging, and fibrosis, that correlate with changes in tissue morphology and matrix composition. Hence, detailed descriptions of the steps involved in sensing and responding to the microenvironment are needed to better understand both the mechanisms of tissue homeostasis and the pathomechanisms of human disease.

On the relation between filament density, force generation, and protrusion rate in mesenchymal cell motility

Lamellipodia are flat membrane protrusions formed during mesenchymal motion. Polymerization at the leading edge assembles the actin filament network and generates protrusion force. How this force is supported by the network and how the assembly rate is shared between protrusion and network retrograde flow determines the protrusion rate. We use mathematical modeling to understand experiments changing the F-actin density in lamellipodia of B16-F1 melanoma cells by modulation of Arp2/3 complex activity or knockout of the formins FMNL2 and FMNL3. Cells respond to a reduction of density with a decrease of protrusion velocity, an increase in the ratio of force to filament number, but constant network assembly rate. The relation between protrusion force and tension gradient in the F-actin network and the density dependency of friction, elasticity, and viscosity of the network explain the experimental observations. The formins act as filament nucleators and elongators with differential rates. Modulation of their activity suggests an effect on network assembly rate. Contrary to these expectations, the effect of changes in elongator composition is much weaker than the consequences of the density change. We conclude that the force acting on the leading edge membrane is the force required to drive F-actin network retrograde flow.

Direct Observation of Single-Molecule Stick-Slip Motion in Polyamide Single Crystals

Stick-slip is a ubiquitous motion in the hydrogen bonding network, which confers the corresponding materials with excellent toughness and strength. The experimental study of the stick-slip mechanism remains challenging because of the complexity of stress accumulation and release. An ideal system for study of this motion should comprise a defined molecular structure and chain arrangement and strong intermolecular interactions. In this study, we detected the stick-slip motion at the single-molecule level in the hydrogen bonding network of polyamide (PA) single crystals through atomic force microscopy (AFM)-based single-molecule force spectroscopy. Our results show that a stiffer force-loading device can enhance the stick capacity by increasing the fracture force and facilitating stress release. We confirm that the chain rotates while slipping and the slip distance is dependent on the unit structure of the hydrogen bonding network.

Buckle up: Membrane tension drives lamellipodial network compression and adhesion deposition

Keren and Shemesh preview work from Pontes et al. that elucidates how the mechanical signals of membrane tension interact with the cytoskeleton to determine adhesion positions.

Whether to spread on a surface or to crawl, cells must apply traction forces to the underlying substrate via adhesion complexes. In this issue, Pontes et al. (2017. J. Cell Biol.https://doi.org/10.1083/jcb.201611117) shed new light on how the interplay among membrane tension, the lamellipodial actin network, and adhesions coordinate the dynamics of spreading fibroblasts.

Membrane tension controls adhesion positioning at the leading edge of cells

Pontes et al. show that plasma membrane mechanics exerts an upstream control during cell motility. Variations in plasma membrane tension orchestrate the behavior of the cell leading edge, with increase–decrease cycles in tension promoting adhesion row positioning.

Cell migration is dependent on adhesion dynamics and actin cytoskeleton remodeling at the leading edge. These events may be physically constrained by the plasma membrane. Here, we show that the mechanical signal produced by an increase in plasma membrane tension triggers the positioning of new rows of adhesions at the leading edge. During protrusion, as membrane tension increases, velocity slows, and the lamellipodium buckles upward in a myosin II–independent manner. The buckling occurs between the front of the lamellipodium, where nascent adhesions are positioned in rows, and the base of the lamellipodium, where a vinculin-dependent clutch couples actin to previously positioned adhesions. As membrane tension decreases, protrusion resumes and buckling disappears, until the next cycle. We propose that the mechanical signal of membrane tension exerts upstream control in mechanotransduction by periodically compressing and relaxing the lamellipodium, leading to the positioning of adhesions at the leading edge of cells.

Physical Model for Stabilization and Repair of Trans-endothelial Apertures

Bacterial toxins that disrupt the stability of contractile structures in endothelial cells promote the opening of large-scale apertures, thereby breaching the endothelium barrier. These apertures are formed by fusion of the basal and apical membranes into a tunnel that spans the height of the cell. Subsequent to the aperture formation, an active repair process, driven by a stimulated polymerization of actin, results in asymmetrical membrane protrusions and, ultimately, the closure of the aperture. Here, we propose a physics-based model for the generation, stabilization and repair of trans-endothelial apertures. Our model is based on the mechanical interplay between tension in the plasma membrane and stresses that develop within different actin structures at the aperture's periphery. We suggest that accumulation of cytoskeletal fragments around the aperture's rim during the expansion phase results in parallel bundles of actin filaments and myosin motors, generating progressively greater contraction forces that resist further expansion of the aperture. Our results indicate that closure of the tunnel is driven by mechanical stresses that develop within a cross-linked actin gel that forms at localized regions of the aperture periphery. We show that stresses within the gel are due to continuous polymerization of actin filaments against the membrane surfaces of the aperture's edges. Based on our mechanical model, we construct a dynamic simulation of the aperture repair process. Our model fully accounts for the phenomenology of the trans-endothelial aperture formation and stabilization, and recaptures the experimentally observed asymmetry of the intermediate aperture shapes during closure. We make experimentally testable predictions for localization of myosin motors to the tunnel periphery and of adhesion complexes to the edges of apertures undergoing closure, and we estimate the minimal nucleation size of cross-linked actin gel that can lead to a successful repair of the aperture.

Shifting the optimal stiffness for cell migration

Cell migration, which is central to many biological processes including wound healing and cancer progression, is sensitive to environmental stiffness, and many cell types exhibit a stiffness optimum, at which migration is maximal. Here we present a cell migration simulator that predicts a stiffness optimum that can be shifted by altering the number of active molecular motors and clutches. This prediction is verified experimentally by comparing cell traction and F-actin retrograde flow for two cell types with differing amounts of active motors and clutches: embryonic chick forebrain neurons (ECFNs; optimum ∼1 kPa) and U251 glioma cells (optimum ∼100 kPa). In addition, the model predicts, and experiments confirm, that the stiffness optimum of U251 glioma cell migration, morphology and F-actin retrograde flow rate can be shifted to lower stiffness by simultaneous drug inhibition of myosin II motors and integrin-mediated adhesions.

Cell migration is sensitive to environmental stiffness, but how cells sense optimal stiffness is not known. Here the authors develop a model that predicts that the optimum can be shifted by altering the number of active molecular motors and clutches, and verify their model in two cell types.

Integrin and cadherin clusters: A robust way to organize adhesions for cell mechanics

Recent studies at the nanometer scale have revealed that relatively uniform clusters of adhesion proteins (50-100 nm) constitute the modular units of cell adhesion sites in both cell-matrix and cell-cell adhesions. Super resolution microscopy and membrane protein diffusion studies both suggest that even large focal adhesions are formed of 100 nm clusters that are loosely aggregated. Clusters of 20-50 adhesion molecules (integrins or cadherins) can support large forces through avidity binding interactions but can also be disassembled or endocytosed rapidly. Assembly of the clusters of integrins is force-independent and involves gathering integrins at ligand binding sites where they are stabilized by cytoplasmic adhesion proteins that crosslink the integrin cytoplasmic tails plus connect the clusters to the cell cytoskeleton. Cooperative-signaling events can occur in a single cluster without cascading to other clusters. Thus, the clusters appear to be very important elements in many cellular processes and can be considered as a critical functional module.

Actin flow and talin dynamics govern rigidity sensing in actin-integrin linkage through talin extension

At cell-substrate adhesion sites, the linkage between actin filaments and integrin is regulated by mechanical stiffness of the substrate. Of potential molecular regulators, the linker proteins talin and vinculin are of particular interest because mechanical extension of talin induces vinculin binding with talin, which reinforces the actin-integrin linkage. For understanding the molecular and biophysical mechanism of rigidity sensing at cell-substrate adhesion sites, we constructed a simple physical model to examine a role of talin extension in the stiffness-dependent regulation of actin-integrin linkage. We show that talin molecules linking between retrograding actin filaments and substrate-bound integrin are extended in a manner dependent on substrate stiffness. The model predicts that, in adhesion complexes containing ≈30 talin links, talin is extended enough for vinculin binding when the substrate is stiffer than 1 kPa. The lifetime of talin links needs to be 2-5 s to achieve an appropriate response of talin extension against substrate stiffness. Furthermore, changes in actin velocity drastically shift the range of substrate stiffness that induces talin-vinculin binding. Our results suggest that talin extension is a key step in sensing and responding to substrate stiffness at cell adhesion sites.

Dynamic reorganization of the actin cytoskeleton

Cellular processes, including morphogenesis, polarization, and motility, rely on a variety of actin-based structures. Although the biochemical composition and filament organization of these structures are different, they often emerge from a common origin. This is possible because the actin structures are highly dynamic. Indeed, they assemble, grow, and disassemble in a time scale of a second to a minute. Therefore, the reorganization of a given actin structure can promote the formation of another. Here, we discuss such transitions and illustrate them with computer simulations.

Two-Phase Acto-Cytosolic Fluid Flow in a Moving Keratocyte: A 2D Continuum Model

The F-actin network and cytosol in the lamellipodia of crawling cells flow in a centripetal pattern and spout-like form, respectively. We have numerically studied this two-phase flow in the realistic geometry of a moving keratocyte. Cytosol has been treated as a low viscosity Newtonian fluid flowing through the high viscosity porous medium of F-actin network. Other involved phenomena including myosin activity, adhesion friction, and interphase interaction are also discussed to provide an overall view of this problem. Adopting a two-phase coupled model by myosin concentration, we have found new accurate perspectives of acto-cytosolic flow and pressure fields, myosin distribution, as well as the distribution of effective forces across the lamellipodia of a keratocyte with stationary shape. The order of magnitude method is also used to determine the contribution of forces in the internal dynamics of lamellipodia.

Integration of actin dynamics and cell adhesion by a three-dimensional, mechanosensitive molecular clutch

During cell migration, the forces generated in the actin cytoskeleton are transmitted across transmembrane receptors to the extracellular matrix or other cells through a series of mechanosensitive, regulable protein-protein interactions termed the molecular clutch. In integrin-based focal adhesions, the proteins forming this linkage are organized into a conserved three-dimensional nano-architecture. Here we discuss how the physical interactions between the actin cytoskeleton and focal-adhesion-associated molecules mediate force transmission from the molecular clutch to the extracellular matrix.

Model for adhesion clutch explains biphasic relationship between actin flow and traction at the cell leading edge

Cell motility relies on the continuous reorganization of a dynamic actin-myosin-adhesion network at the leading edge of the cell, in order to generate protrusion at the leading edge and traction between the cell and its external environment. We analyze experimentally measured spatial distributions of actin flow, traction force, myosin density, and adhesion density in control and pharmacologically perturbed epithelial cells in order to develop a mechanical model of the actin-adhesion-myosin self-organization at the leading edge. A model in which the F-actin network is treated as a viscous gel, and adhesion clutch engagement is strengthened by myosin but weakened by actin flow, can explain the measured molecular distributions and correctly predict the spatial distributions of the actin flow and traction stress. We test the model by comparing its predictions with measurements of the actin flow and traction stress in cells with fast and slow actin polymerization rates. The model predicts how the location of the lamellipodium-lamellum boundary depends on the actin viscosity and adhesion strength. The model further predicts that the location of the lamellipodium-lamellum boundary is not very sensitive to the level of myosin contraction.

Active mechanical coupling between the nucleus, cytoskeleton and the extracellular matrix, and the implications for perinuclear actomyosin organization

Experimental and theoretical studies have demonstrated that the polarization of actomyosin forces in the cytoskeleton of adherent cells is governed by local elastic stresses. Based on this phenomenon, and the established observation that the nucleus is mechanically connected to the extracellular matrix (ECM) via the cytoskeleton, we theoretically analyze here the active mechanical coupling between the nucleus, cytoskeleton and the ECM. The cell is modeled as an active spherical inclusion, containing a round nucleus at its center, and embedded in a 3D elastic matrix. We investigate three sources of cellular stress: spreading-induced stress, actomyosin contractility and chromatin entropic forces. Formulating the coupling of actomyosin contractility to the local stress we predict the consequences that the nucleus, cytoskeleton and ECM mechanical properties may have on the overall force-balance in the cell and the perinuclear acto-myosin polarization. We demonstrate that the presence of the nucleus induces symmetry breaking of the elastic stress that, we predict, elastically tends to orient actomyosin alignment tangentially around the nucleus; the softer the nucleus or the matrix, the stronger is the preference for tangential alignment. Spreading induced stresses may induce radial actomyosin alignment near stiff nuclei. In addition, we show that in regions of high actomyosin density myosin motors have an elastic tendency to orient tangentially as often occurs near the cell periphery. These conclusions highlight the role of the nucleus in the regulation of cytoskeleton organization and may provide new insight into the mechanics of stem cell differentiation involving few fold increase in nucleus stiffness.

Dynamics of cell shape and forces on micropatterned substrates predicted by a cellular Potts model

Micropatterned substrates are often used to standardize cell experiments and to quantitatively study the relation between cell shape and function. Moreover, they are increasingly used in combination with traction force microscopy on soft elastic substrates. To predict the dynamics and steady states of cell shape and forces without any a priori knowledge of how the cell will spread on a given micropattern, here we extend earlier formulations of the two-dimensional cellular Potts model. The third dimension is treated as an area reservoir for spreading. To account for local contour reinforcement by peripheral bundles, we augment the cellular Potts model by elements of the tension-elasticity model. We first parameterize our model and show that it accounts for momentum conservation. We then demonstrate that it is in good agreement with experimental data for shape, spreading dynamics, and traction force patterns of cells on micropatterned substrates. We finally predict shapes and forces for micropatterns that have not yet been experimentally studied.

Determinants of maximal force transmission in a motor-clutch model of cell traction in a compliant microenvironment

The mechanical stiffness of a cell's environment exerts a strong, but variable, influence on cell behavior and fate. For example, different cell types cultured on compliant substrates have opposite trends of cell migration and traction as a function of substrate stiffness. Here, we describe how a motor-clutch model of cell traction, which exhibits a maximum in traction force with respect to substrate stiffness, may provide a mechanistic basis for understanding how cells are tuned to sense the stiffness of specific microenvironments. We find that the optimal stiffness is generally more sensitive to clutch parameters than to motor parameters, but that single parameter changes are generally only effective over a small range of values. By contrast, dual parameter changes, such as coordinately increasing the numbers of both motors and clutches offer a larger dynamic range for tuning the optimum. The model exhibits distinct regimes: at high substrate stiffness, clutches quickly build force and fail (so-called frictional slippage), whereas at low substrate stiffness, clutches fail spontaneously before the motors can load the substrate appreciably (a second regime of frictional slippage). Between the two extremes, we find the maximum traction force, which occurs when the substrate load-and-fail cycle time equals the expected time for all clutches to bind. At this stiffness, clutches are used to their fullest extent, and motors are therefore resisted to their fullest extent. The analysis suggests that coordinate parameter shifts, such as increasing the numbers of motors and clutches, could underlie tumor progression and collective cell migration.

Actin dynamics, architecture, and mechanics in cell motility

Tight coupling between biochemical and mechanical properties of the actin cytoskeleton drives a large range of cellular processes including polarity establishment, morphogenesis, and motility. This is possible because actin filaments are semi-flexible polymers that, in conjunction with the molecular motor myosin, can act as biological active springs or "dashpots" (in laymen's terms, shock absorbers or fluidizers) able to exert or resist against force in a cellular environment. To modulate their mechanical properties, actin filaments can organize into a variety of architectures generating a diversity of cellular organizations including branched or crosslinked networks in the lamellipodium, parallel bundles in filopodia, and antiparallel structures in contractile fibers. In this review we describe the feedback loop between biochemical and mechanical properties of actin organization at the molecular level in vitro, then we integrate this knowledge into our current understanding of cellular actin organization and its physiological roles.

Analytical solutions of actin-retrograde-flow in a circular stationary cell: a mechanical point of view

The network of actin filaments in the lamellipodium (LP) of stationary and migrating cells flows in a retrograde direction, from the membrane periphery toward the cell nucleus. We have theoretically studied this phenomenon in the circular stationary (fully spread) cells. Adopting a continuum view on the LP actin network, new closed-form solutions are provided for the actin-retrograde-flow (ARF) in a polar coordinate system. Due to discrepancy in the mechanical models of the actin network in the ARF regime, solutions are provided for both assumptions of solid and fluid behavior. Other involved phenomena, including polymerizing machine at the membrane periphery, cytosol drag, adhesion friction, and membrane tension, are also discussed to provide an overall quantitative view on this problem.

Formation of transient lamellipodia

Cell motility driven by actin polymerization is pivotal to the development and survival of organisms and individual cells. Motile cells plated on flat substrates form membrane protrusions called lamellipodia. The protrusions repeatedly appear and retract in all directions. If a lamellipodium is stabilized and lasts for some time, it can take over the lead and determine the direction of cell motion. Protrusions traveling along the cell perimeter have also been observed. Their initiation is in some situations the effect of the dynamics of the pathway linking plasma membrane receptors to actin filament nucleation, e.g. in chemotaxis. However, lamellipodia are also formed in many cells incessantly during motion with a constant state of the signaling pathways upstream from nucleation promoting factors (NPFs), or spontaneously in resting cells. These observations strongly suggest protrusion formation can also be a consequence of the dynamics downstream from NPFs, with signaling setting the dynamic regime but not initiating the formation of individual protrusions. A quantitative mechanism for this kind of lamellipodium dynamics has not been suggested yet. Here, we present a model exhibiting excitable actin network dynamics. Individual lamellipodia form due to random supercritical filament nucleation events amplified by autocatalytic branching. They last for about 30 seconds to many minutes and are terminated by filament bundling, severing and capping. We show the relevance of the model mechanism for experimentally observed protrusion dynamics by reproducing in very good approximation the repetitive protrusion formation measured by Burnette et al. with respect to the velocities of leading edge protrusion and retrograde flow, oscillation amplitudes, periods and shape, as well as the phase relation between protrusion and retrograde flow. Our modeling results agree with the mechanism of actin bundle formation during lamellipodium retraction suggested by Burnette et al. and Koestler et al.

Michael Kozlov: a twist in membrane physics. Interview by Caitlin Sedwick

Kozlov studies the biophysics of membrane shape and behavior.

Morphogenesis can be driven by properly parametrised mechanical feedback

A fundamental problem of morphogenesis is whether it presents itself as a succession of links that are each driven by its own specific cause-effect relationship, or whether all of the links can be embraced by a common law that is possible to formulate in physical terms. We suggest that a common biophysical background for most, if not all, morphogenetic processes is based upon feedback between mechanical stresses (MS) that are imposed to a given part of a developing embryo by its other parts and MS that are actively generated within that part. The latter are directed toward hyper-restoration (restoration with an overshoot) of the initial MS values. We show that under mechanical constraints imposed by other parts, these tendencies drive forth development. To provide specificity for morphogenetic reactions, this feedback should be modulated by long-term parameters and/or initial conditions that are set up by genetic factors. The experimental and model data related to this concept are reviewed.

Bidirectional coupling between integrin-mediated signaling and actomyosin mechanics explains matrix-dependent intermittency of leading-edge motility

A physicochemical model is used to describe the coupling of adhesion, cytoskeletal, and signaling dynamics during cell migration. Analysis of stochastic simulations predicts relationships between measurable quantities that reflect partitioning of stress between F-actin–bound adhesions, which act as a molecular clutch, and retrograde F-actin flow.

Animal cell migration is a complex process characterized by the coupling of adhesion, cytoskeletal, and signaling dynamics. Here we model local protrusion of the cell edge as a function of the load-bearing properties of integrin-based adhesions, actin polymerization fostered by adhesion-mediated signaling, and mechanosensitive activation of RhoA that promotes myosin II–generated stress on the lamellipodial F-actin network. Analysis of stochastic model simulations illustrates how these pleiotropic functions of nascent adhesions may be integrated to govern temporal persistence and frequency of protrusions. The simulations give mechanistic insight into the documented effects of extracellular matrix density and myosin abundance, and they show characteristic, nonnormal distributions of protrusion duration times that are similar to those extracted from live-cell imaging experiments. Analysis of the model further predicts relationships between measurable quantities that reflect the partitioning of stress between tension on F-actin–bound adhesions, which act as a molecular clutch, and dissipation by retrograde F-actin flow.

Master equation-based analysis of a motor-clutch model for cell traction force

Microenvironmental mechanics play an important role in determining the morphology, traction, migration, proliferation, and differentiation of cells. A stochastic motor-clutch model has been proposed to describe this stiffness sensitivity. In this work, we present a master equation-based ordinary differential equation (ODE) description of the motor-clutch model, from which we derive an analytical expression to for a cell's optimum stiffness (i.e. the stiffness at which the traction force is maximal). This analytical expression provides insight into the requirements for stiffness sensing by establishing fundamental relationships between the key controlling cell-specific parameters. We find that the fundamental controlling parameters are the numbers of motors and clutches (constrained to be nearly equal), and the time scale of the on-off kinetics of the clutches (constrained to favor clutch binding over clutch unbinding). Both the ODE solution and the analytical expression show good agreement with Monte Carlo motor-clutch output, and reduce computation time by several orders of magnitude, which potentially enables long time scale behaviors (hours-days) to be studied computationally in an efficient manner. The ODE solution and the analytical expression may be incorporated into larger scale models of cellular behavior to bridge the gap from molecular time scales to cellular and tissue time scales.

Mathematical modeling of eukaryotic cell migration: insights beyond experiments

A migrating cell is a molecular machine made of tens of thousands of short-lived and interacting parts. Understanding migration means understanding the self-organization of these parts into a system of functional units. This task is one of tackling complexity: First, the system integrates numerous chemical and mechanical component processes. Second, these processes are connected in feedback interactions and over a large range of spatial and temporal scales. Third, many processes are stochastic, which leads to heterogeneous migration behaviors. Early on in the research of cell migration it became evident that this complexity exceeds human intuition. Thus, the cell migration community has led the charge to build mathematical models that could integrate the diverse experimental observations and measurements in consistent frameworks, first in conceptual and more recently in molecularly explicit models. The main goal of this review is to sift through a series of important conceptual and explicit mathematical models of cell migration and to evaluate their contribution to the field in their ability to integrate critical experimental data.

Effects of adhesion dynamics and substrate compliance on the shape and motility of crawling cells

Computational modeling of eukaryotic cells moving on substrates is an extraordinarily complex task: many physical processes, such as actin polymerization, action of motors, formation of adhesive contacts concomitant with both substrate deformation and recruitment of actin etc., as well as regulatory pathways are intertwined. Moreover, highly nontrivial cell responses emerge when the substrate becomes deformable and/or heterogeneous. Here we extended a computational model for motile cell fragments, based on an earlier developed phase field approach, to account for explicit dynamics of adhesion site formation, as well as for substrate compliance via an effective elastic spring. Our model displays steady motion vs. stick-slip transitions with concomitant shape oscillations as a function of the actin protrusion rate, the substrate stiffness, and the rates of adhesion. Implementing a step in the substrate's elastic modulus, as well as periodic patterned surfaces exemplified by alternating stripes of high and low adhesiveness, we were able to reproduce the correct motility modes and shape phenomenology found experimentally. We also predict the following nontrivial behavior: the direction of motion of cells can switch from parallel to perpendicular to the stripes as a function of both the adhesion strength and the width ratio of adhesive to non-adhesive stripes.

A kinetic model for RNA-interference of focal adhesions

BACKGROUND

Focal adhesions are integrin-based cell-matrix contacts that transduce and integrate mechanical and biochemical cues from the environment. They develop from smaller and more numerous focal complexes under the influence of mechanical force and are key elements for many physiological and disease-related processes, including wound healing and metastasis. More than 150 different proteins localize to focal adhesions and have been systematically classified in the adhesome project (http://www.adhesome.org). First RNAi-screens have been performed for focal adhesions and the effect of knockdown of many of these components on the number, size, shape and location of focal adhesions has been reported.

RESULTS

We have developed a kinetic model for RNA interference of focal adhesions which represents some of its main elements: a spatially layered structure, signaling through the small GTPases Rac and Rho, and maturation from focal complexes to focal adhesions under force. The response to force is described by two complementary scenarios corresponding to slip and catch bond behavior, respectively. Using estimated and literature values for the model parameters, three time scales of the dynamics of RNAi-influenced focal adhesions are identified: a sub-minute time scale for the assembly of focal complexes, a sub-hour time scale for the maturation to focal adhesions, and a time scale of days that controls the siRNA-mediated knockdown. Our model shows bistability between states dominated by focal complexes and focal adhesions, respectively. Catch bonding strongly extends the range of stability of the state dominated by focal adhesions. A sensitivity analysis predicts that knockdown of focal adhesion components is more efficient for focal adhesions with slip bonds or if the system is in a state dominated by focal complexes. Knockdown of Rho leads to an increase of focal complexes.

CONCLUSIONS

The suggested model provides a kinetic description of the effect of RNA-interference of focal adhesions. Its predictions are in good agreement with known experimental results and can now guide the design of RNAi-experiments. In the future, it can be extended to include more components of the adhesome. It also could be extended by spatial aspects, for example by the differential activation of the Rac- and Rho-pathways in different parts of the cell.