Modeling of the actin-cytoskeleton in symmetric lamellipodial fragments

The pushing structures of cells include laminar sheets, termed lamellipodia, made up of a meshwork of actin filaments that grow at the front and depolymerise at the rear, in a treadmilling mode. We here develop a mathematical model to describe the turnover and the mechanical properties of this network.Our basic modeling assumptions are that the lamellipodium is idealised as a two-dimensional structure, and that the actin network consists of two families of possibly bent, but locally parallel filaments. Instead of dealing with individual polymers, the filaments are assumed to be continuously distributed.The model includes (de)polymerization, of the mechanical effects of cross-linking, cell-substrate adhesion, as well as of the leading edge of the membrane.In the first version presented here, the total amount of F-actin is prescribed by assuming a constant polymerisation speed at the leading edge and a fixed total number and length distribution of filaments. We assume that cross-links at filament crossing points as well as integrin linkages with the matrix break and reform in response to incremental changes in network organization. In this first treatment, the model successfully simulates the persistence of the treadmilling network in radially spread cells.Key words: modelling, cell movement, actin-network     

The role of actin turnover in retrograde actin network flow in neuronal growth cones

The balance of actin filament polymerization and depolymerization maintains a steady state network treadmill in neuronal growth cones essential for motility and guidance. Here we have investigated the connection between depolymerization and treadmilling dynamics. We show that polymerization-competent barbed ends are concentrated at the leading edge and depolymerization is distributed throughout the peripheral domain. We found a high-to-low G-actin gradient between peripheral and central domains. Inhibiting turnover with jasplakinolide collapsed this gradient and lowered leading edge barbed end density. Ultrastructural analysis showed dramatic reduction of leading edge actin filament density and filament accumulation in central regions. Live cell imaging revealed that the leading edge retracted even as retrograde actin flow rate decreased exponentially. Inhibition of myosin II activity before jasplakinolide treatment lowered baseline retrograde flow rates and prevented leading edge retraction. Myosin II activity preferentially affected filopodial bundle disassembly distinct from the global effects of jasplakinolide on network turnover. We propose that growth cone retraction following turnover inhibition resulted from the persistence of myosin II contractility even as leading edge assembly rates decreased. The buildup of actin filaments in central regions combined with monomer depletion and reduced polymerization from barbed ends suggests a mechanism for the observed exponential decay in actin retrograde flow. Our results show that growth cone motility is critically dependent on continuous disassembly of the peripheral actin network.     

A mechanistic model of the actin cycle

We have derived a broad, deterministic model of the steady-state actin cycle that includes its major regulatory mechanisms. Ours is the first model to solve the complete nucleotide profile within filaments, a feature that determines the dynamics and geometry of actin networks at the leading edges of motile cells, and one that has challenged investigators developing models to interpret steady-state experiments. We arrived at the nucleotide profile through analytic and numerical approaches that completely agree. Our model reproduces behaviors seen in numerous experiments with purified proteins, but allows a detailed inspection of the concentrations and fluxes that might exist in these experiments. These inspections provide new insight into the mechanisms that determine the rate of actin filament treadmilling. Specifically, we find that mechanisms for enhancing Pi release from the ADP.Pi intermediate on filaments, for increasing the off rate of ADP-bound subunits at pointed ends, and the multiple, simultaneous functions of profilin, make unique and essential contributions to increased treadmilling. In combination, these mechanisms have a theoretical capacity to increase treadmilling to levels limited only by the amount of available actin. This limitation arises because as the cycle becomes more dynamic, it tends toward the unpolymerized state.     

Arp2/3 complex and actin depolymerizing factor/cofilin in dendritic organization and treadmilling of actin filament array in lamellipodia.

The leading edge (approximately 1 microgram) of lamellipodia in Xenopus laevis keratocytes and fibroblasts was shown to have an extensively branched organization of actin filaments, which we term the dendritic brush. Pointed ends of individual filaments were located at Y-junctions, where the Arp2/3 complex was also localized, suggesting a role of the Arp2/3 complex in branch formation. Differential depolymerization experiments suggested that the Arp2/3 complex also provided protection of pointed ends from depolymerization. Actin depolymerizing factor (ADF)/cofilin was excluded from the distal 0.4 micrometer++ of the lamellipodial network of keratocytes and in fibroblasts it was located within the depolymerization-resistant zone. These results suggest that ADF/cofilin, per se, is not sufficient for actin brush depolymerization and a regulatory step is required. Our evidence supports a dendritic nucleation model (Mullins, R.D., J.A. Heuser, and T.D. Pollard. 1998. Proc. Natl. Acad. Sci. USA. 95:6181-6186) for lamellipodial protrusion, which involves treadmilling of a branched actin array instead of treadmilling of individual filaments. In this model, Arp2/3 complex and ADF/cofilin have antagonistic activities. Arp2/3 complex is responsible for integration of nascent actin filaments into the actin network at the cell front and stabilizing pointed ends from depolymerization, while ADF/cofilin promotes filament disassembly at the rear of the brush, presumably by pointed end depolymerization after dissociation of the Arp2/3 complex.     

Exchange of actin subunits at the leading edge of living fibroblasts: possible role of treadmilling

Previous observations indicated that the lamellipodium ("leading edge") of fibroblasts contains a dense meshwork, as well as numerous bundles (microspikes) of actin filaments. Most, if not all, of the filaments have a uniform polarity, with the "barbed" end associated with the membrane. I investigated whether and how actin subunits exchange in this region by microinjecting living gerbil fibroma cells (IMR-33) with actin that had been labeled with iodoacetamidotetramethylrhodamine. After incorporation of the labeled actin into the lamellipodium, I used a laser microbeam to photobleach a 3-4-micron region at and surrounding a microspike, without disrupting the integrity of the structure. I then recorded the pattern of fluorescence recovery and analyzed it using a combination of TV image intensification and digital image processing techniques. Fluorescence recovery was first detected near the edge of the cell and then moved toward the cell's center at a constant rate of 0.79 +/- 0.31 micron/min. When only part of the lamellipodium near the edge of the cell was photobleached, the bleached spot also moved toward the cell's center and through an area unbleached by the laser beam. These results indicated that steady state incorporation of actin subunits occurred predominantly at the membrane-associated end of actin filaments, and that actin subunits in the lamellipodium underwent a constant movement toward the center of the cell. I suggest that treadmilling, possibly in combination with other molecular interactions, may provide an effective mechanism for the movement of actin subunits and the protrusion of cytoplasm in the lamellipodium of fibroblasts.     

Stress-induced alignment of actin filaments and the mechanics of cytogel

Experimental evidence suggests that anisotropic stress induces alignment of intracellular actin filaments. We develop a model for this phenomenon, which includes a parameter reflecting the sensitivity of the microfilament network to changes in the stress field. When applied to a uniform cell sheet at rest, the model predicts that for sufficiently large values of the sensitivity parameter, all the actin filaments will spontaneously align in a single direction. Stress alignment can also be caused by a change in external conditions, and as an example of this we apply our model to the initial response of embryonic epidermis to wounding. Our solutions in this case are able to reflect the actin cable that has been found at the wound edge in recent experiments; the cable consists of microfilaments aligned with stress at the wound boundary of the epithelium. These applications suggest that stress-induced alignment of actin filaments could play a key role in some biological systems. This is the first attempt to include the alignment phenomenon in a mechanical model of cytogel.     

Implications of treadmilling for the stability and polarity of actin and tubulin polymers in vivo

In this report, we examine how the cell can selectively stabilize anchored filaments and suppress spontaneous filament assembly. Because microtubules and actin filaments have an organized distribution in cells, the cell must have a mechanism for suppressing spontaneous and random polymerization. Though the mechanism for suppressing spontaneous polymerization is unknown, an unusual property of these filaments has been demonstrated recently, i.e., under steady-stae conditions, in vitro actin filaments and microtubules can exhibit a flux of subunits through the polymers called "treadmilling." In vivo, however, most, if not all, of these polymers are attached at one end to specific structures and treadmilling should not occur. The function of treadmilling in vivo is, therefore, unclear at present. However, as shown here, the same physicochemical property of coupling assembly to ATP or GTP hydrolysis that leads to treadmilling in vitro can act to selectively stabilize anchored polymers in vivo. I show here that the theory of treadmilling implies that the concentration of subunits necessary for assembly of the nonanchored polymer will in general be higher than the concentration necessary for the assembly of polymers anchored with a specific polarity. This disparity in the monomer concentrations required for assembly can lead to a selective stabilization of anchored polymers and complete suppression of spontaneous polymerization at apparent equilibrium in vivo. It is possible, therefore, that the phenomenon of treadmilling is an in vitro manifestation of a mechanism designed to use ATP or GTP hydrolysis to control the spatial organization of filaments in the cell.     

Moving and stationary actin filaments are involved in spreading of postmitotic PtK2 cells

We have investigated spreading of postmitotic PtK2 cells and the behavior of actin filaments in this system by time-lapse microscopy and photoactivation of fluorescence. During mitosis PtK2 cells round up and at cytokinesis the daughter cells spread back to regain their interphase morphology. Normal spreading edges are quite homogenous and are not comprised of two distinct areas (lamellae and lamellipodia) as found in moving edges of interphase motile cells. Spreading edges are connected to a network of long, thin, actin-rich fibers called retraction fibers. A role for retraction fibers in spreading was tested by mechanical disruption of fibers ahead of a spreading edge. Spreading is inhibited over the region of disruption, but not over neighboring intact fibers. Using photoactivation of fluorescence to mark actin filaments, we have determined that the majority of actin filaments move forward in spreading edges at the same rate as the edge. As far as we are aware, this is the first time that forward movement of a cell edge has been correlated with forward movement of actin filaments. In contrast, actin filaments in retraction fibers remain stationary with respect to the substrate. Thus there are at least two dynamic populations of actin polymer in spreading postmitotic cells. This is supported by the observation that actin filaments in some spreading edges not only move forward, but also separate into two fractions or broaden with time. A small fraction of postmitotic cells have a spreading edge with a distinct lamellipodium. In these edges, marked actin polymer fluxes backward with respect to substrate. We suggest that forward movement of actin filaments may participate in generating force for spreading in postmitotic cells and perhaps more generally for cell locomotion.     

Filamin A, the Arp2/3 complex, and the morphology and function of cortical actin filaments in human melanoma cells.

The Arp2/3 complex and filamin A (FLNa) branch actin filaments. To define the role of these actin-binding proteins in cellular actin architecture, we compared the morphology of FLNa-deficient human melanoma (M2) cells and three stable derivatives of these cells expressing normal FLNa concentrations. All the cell lines contain similar amounts of the Arp2/3 complex. Serum addition causes serum-starved M2 cells to extend flat protrusions transiently; thereafter, the protrusions turn into spherical blebs and the cells do not crawl. The short-lived lamellae of M2 cells contain a dense mat of long actin filaments in contrast to a more three-dimensional orthogonal network of shorter actin filaments in lamellae of identically treated FLNa-expressing cells capable of translational locomotion. FLNa-specific antibodies localize throughout the leading lamellae of these cells at junctions between orthogonally intersecting actin filaments. Arp2/3 complex-specific antibodies stain diffusely and label a few, although not the same, actin filament overlap sites as FLNa antibody. We conclude that FLNa is essential in cells that express it for stabilizing orthogonal actin networks suitable for locomotion. Contrary to some proposals, Arp2/3 complex-mediated branching of actin alone is insufficient for establishing an orthogonal actin organization or maintaining mechanical stability at the leading edge.     

The coordination between actin filaments and adhesion in mesenchymal migration

Mesenchymal cell motility is characterized by a polarized distribution of actin filaments, with a network of short branched actin filaments at the leading edge, and polymers of actin filaments arranged into distinct classes of actin stress fibres behind the leading edge. Importantly, the distinct actin filaments are characteristically associated with discrete adhesion structures and both the adhesions and the actin filaments are co-ordinately regulated during cell migration. While it has long been known that these macromolecular structures are intimately linked in cells, precisely how they are co-ordinately regulated is presently unknown. Live imaging data now suggests that the focal adhesions may act as sites of actin polymerization resulting in the generation of tension-bearing actin bundles of actin filaments (stress fibres). Moreover, a picture is emerging to suggest that the tropomyosin family of proteins that can determine actin filament dynamics may also play a key role in determining the transition between adhesion states. Molecules such as the tropomyosins are therefore tantalizing candidates to orchestrate the coordination of actin and adhesion dynamics during mesenchymal cell migration.     

The physics of filopodial protrusion

Filopodium, a spike-like actin protrusion at the leading edge of migrating cells, functions as a sensor of the local environment and has a mechanical role in protrusion. We use modeling to examine mechanics and spatial-temporal dynamics of filopodia. We find that >10 actin filaments have to be bundled to overcome the membrane resistance and that the filopodial length is limited by buckling for 10-30 filaments and by G-actin diffusion for >30 filaments. There is an optimal number of bundled filaments, approximately 30, at which the filopodial length can reach a few microns. The model explains characteristic interfilopodial distance of a few microns as a balance of initiation, lateral drift, and merging of the filopodia. The theory suggests that F-actin barbed ends have to be focused and protected from capping (the capping rate has to decrease one order of magnitude) once every hundred seconds per micron of the leading edge to initiate the observed number of filopodia. The model generates testable predictions about how filopodial length, rate of growth, and interfilopodial distance should depend on the number of bundled filaments, membrane resistance, lamellipodial protrusion rate, and G-actin diffusion coefficient.     

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.     

Inhibition of the Arp2/3 complex-nucleated actin polymerization and branch formation by tropomyosin

The actin filament network immediately under the plasma membrane at the leading edge of rapidly moving cells consists of short, branched filaments, while those deeper in the cortex are much longer and are rarely branched. Nucleation by the Arp2/3 complex activated by membrane-bound factors (Rho-family GTPases and PIP(2)) is postulated to account for the formation of the branched network. Tropomyosin (TM) binds along the sides of filaments and protects them from severing proteins and pointed-end depolymerization in vitro. Here, we show that TM inhibits actin filament branching and nucleation by the Arp2/3 complex activated by WASp-WA. Tropomyosin increases the lag at the outset of polymerization, reduces the concentration of ends by 75%, and reduces the number of branches by approximately 50%. We conclude that TM bound to actin filaments inhibits their ability to act as secondary activators of nucleation by the Arp2/3 complex. This is the first example of inhibition of branching by an actin binding protein. We suggest that TM suppresses the nucleation of actin filament branches from actin filaments in the deep cortex of motile cells. Other abundant actin binding proteins may also locally regulate the branching nucleation by the Arp2/3 complex in cells.     

Nucleation causes an actin network to fragment into multiple high-density domains

Actin networks rely on nucleation mechanisms to generate new filaments because spontaneous nucleation is kinetically disfavored. Branching nucleation of actin filaments by actin-related protein (Arp2/3), in particular, is critical for actin self-organization. In this study, we use the simulation platform for active matter MEDYAN to generate 2000 s long stochastic trajectories of actin networks, under varying Arp2/3 concentrations, in reaction volumes of biologically meaningful size (>20 μm³). We find that the dynamics of Arp2/3 increase the abundance of short filaments and increases network treadmilling rate. By analyzing the density fields of F-actin, we find that at low Arp2/3 concentrations, F-actin is organized into a single connected and contractile domain, while at elevated Arp2/3 levels (10 nM and above), such high-density actin domains fragment into smaller domains spanning a wide range of volumes. These fragmented domains are extremely dynamic, continuously merging and splitting, owing to the high treadmilling rate of the underlying actin network. Treating the domain dynamics as a drift-diffusion process, we find that the fragmented state is stochastically favored, and the network state slowly drifts toward the fragmented state with considerable diffusion (variability) in the number of domains. We suggest that tuning the Arp2/3 concentration enables cells to transition from a globally coherent cytoskeleton, whose response involves the entire cytoplasmic network, to a fragmented cytoskeleton, where domains can respond independently to locally varying signals.     

Attachment conditions control actin filament buckling and the production of forces

Actin polymerization is the driving force for a large number of cellular processes. Formation of lamellipodia and filopodia at the leading edge of motile cells requires actin polymerization induced mechanical deformation of the plasma membrane. To generate different types of membrane protrusions, the mechanical properties of actin filaments can be constrained by interacting proteins. A striking example of such constraint is the buckling of actin filaments generated in vitro by the cooperative effect of a processive actin nucleating factor (formin) and a molecular motor (myosin II). We developed a physical model based on equations for an elastic rod that accounts for actin filament buckling. Both ends of the rod were maintained in a fixed position in space and we considered three sets of boundary conditions. The model qualitatively and quantitatively reproduces the shape distribution of actin filaments. We found that actin polymerization counterpoises a force in the range 0.4-1.6 pN for moderate end-to-end distance (approximately 1 microm) and could be as large as 10 pN for shorter distances. If the actin rod attachment includes a spring, we discovered that the stiffness must be in the range 0.1-1.2 pN/nm to account for the observed buckling.     

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.     

Robust organizational principles of protrusive biopolymer networks in migrating living cells

Cell migration is associated with the dynamic protrusion of a thin actin-based cytoskeletal extension at the cell front, which has been shown to consist of two different substructures, the leading lamellipodium and the subsequent lamellum. While the formation of the lamellipodium is increasingly well understood, organizational principles underlying the emergence of the lamellum are just beginning to be unraveled. We report here on a 1D mathematical model which describes the reaction-diffusion processes of a polarized actin network in steady state, and reproduces essential characteristics of the lamellipodium-lamellum system. We observe a steep gradient in filament lengths at the protruding edge, a local depolymerization maximum a few microns behind the edge, as well as a differential dominance of the network destabilizer ADF/cofilin and the stabilizer tropomyosin. We identify simple and robust organizational principles giving rise to the derived network characteristics, uncoupled from the specifics of any molecular implementation, and thus plausibly valid across cell types. An analysis of network length dependence on physico-chemical system parameters implies that to limit array treadmilling to cellular dimensions, network growth has to be truncated by mechanisms other than aging-induced depolymerization, e.g., by myosin-associated network dissociation at the transition to the cell body. Our work contributes to the analytical understanding of the cytoskeletal extension's bisection into lamellipodium and lamellum and sheds light on how cells organize their molecular machinery to achieve motility.     

Nerve growth cone lamellipodia contain two populations of actin filaments that differ in organization and polarity

The organization and polarity of actin filaments in neuronal growth cones was studied with negative stain and freeze-etch EM using a permeabilization protocol that caused little detectable change in morphology when cultured nerve growth cones were observed by video-enhanced differential interference contrast microscopy. The lamellipodial actin cytoskeleton was composed of two distinct subpopulations: a population of 40-100-nm-wide filament bundles radiated from the leading edge, and a second population of branching short filaments filled the volume between the dorsal and ventral membrane surfaces. Together, the two populations formed the three-dimensional structural network seen within expanding lamellipodia. Interaction of the actin filaments with the ventral membrane surface occurred along the length of the filaments via membrane associated proteins. The long bundled filament population was primarily involved in these interactions. The filament tips of either population appeared to interact with the membrane only at the leading edge; this interaction was mediated by a globular Triton-insoluble material. Actin filament polarity was determined by decoration with myosin S1 or heavy meromyosin. Previous reports have suggested that the polarity of the actin filaments in motile cells is uniform, with the barbed ends toward the leading edge. We observed that the actin filament polarity within growth cone lamellipodia is not uniform; although the predominant orientation was with the barbed end toward the leading edge (47-56%), 22-25% of the filaments had the opposite orientation with their pointed ends toward the leading edge, and 19-31% ran parallel to the leading edge. The two actin filament populations display distinct polarity profiles: the longer filaments appear to be oriented predominantly with their barbed ends toward the leading edge, whereas the short filaments appear to be randomly oriented. The different length, organization and polarity of the two filament populations suggest that they differ in stability and function. The population of bundled long filaments, which appeared to be more ventrally located and in contact with membrane proteins, may be more stable than the population of short branched filaments. The location, organization, and polarity of the long bundled filaments suggest that they may be necessary for the expansion of lamellipodia and for the production of tension mediated by receptors to substrate adhesion molecules.