A multi-modular tensegrity model of an actin stress fiber

Stress fibers are contractile bundles in the cytoskeleton that stabilize cell structure by exerting traction forces on the extracellular matrix. Individual stress fibers are molecular bundles composed of parallel actin and myosin filaments linked by various actin-binding proteins, which are organized end-on-end in a sarcomere-like pattern within an elongated three-dimensional network. While measurements of single stress fibers in living cells show that they behave like tensed viscoelastic fibers, precisely how this mechanical behavior arises from this complex supramolecular arrangement of protein components remains unclear. Here we show that computationally modeling a stress fiber as a multi-modular tensegrity network can predict several key behaviors of stress fibers measured in living cells, including viscoelastic retraction, fiber splaying after severing, non-uniform contraction, and elliptical strain of a puncture wound within the fiber. The tensegrity model can also explain how they simultaneously experience passive tension and generate active contraction forces; in contrast, a tensed cable net model predicts some, but not all, of these properties. Thus, tensegrity models may provide a useful link between molecular and cellular scale mechanical behaviors and represent a new handle on multi-scale modeling of living materials.     

The role of prestress and architecture of the cytoskeleton and deformability of cytoskeletal filaments in mechanics of adherent cells: a quantitative analysis.

Mechanical properties of adherent cells were investigated using methods of engineering mechanics. The cytoskeleton (CSK) was modeled as a filamentous network and key mechanisms and corresponding molecular structures which determine cell elastic behavior were identified. Three models of the CSK were considered: open-cell foam networks, prestressed cable nets, and a tensegrity model as a special case of the latter. For each model, the modulus of elasticity (i.e. an index of resistance to small deformation) was given as a function of mechanical and geometrical properties of CSK filaments whose values were determined from the data in the literature. Quantitative predictions for the elastic modulus were compared with data obtained previously from mechanical tests on adherent cells. The open-cell foam model yielded the elastic modulus (10(3)-10(4)Pa) which was consistent with measurements which apply a large compressive stress to the cell. This suggests that bending of CSK filaments is the key mechanism for resisting large compression. The prestressed cable net and tensegrity model yielded much lower elastic moduli (10(1)-10(2)Pa) which were consistent with values determined from equilibrium measurements at low applied stress. This suggests that CSK prestress and architecture are the primary determinants of the cell elastic response. The tensegrity model revealed the possibility that buckling of microtubules of the CSK also contributed to cell elasticity.     

Quantitative analysis of extension-torsion coupling of actin filaments

Actin filaments have a double-helix structure consisting of globular actin molecules. In many mechanical cellular activities, such as cell movement, division, and shape control, modulation of the extensional and torsional dynamics of the filament has been linked to regulatory actin-binding protein functions. Therefore, it is important to quantitatively evaluate extension-torsion coupling of filament to better understand the actin filament dynamics. In the present study, the extension-torsion coupling was investigated using molecular dynamics simulations. We constructed a model for the actin filament consisting of 14 actin subunits in an ionic solvent as a minimal functional unit, and analyzed longitudinal and twisting Brownian motions of the filament. We then derived the expected value of energy associated with extension and torsion at equilibrium, and evaluated the extension-torsion stiffness of the filament from the thermal fluctuations obtained from the MD simulations. The results demonstrated that as the analyzed sampling-window duration was increased, the extension-torsion coupling stiffness evaluated on a nanosecond scale tended to converge to a value of 7.6×10(-11) N. The results obtained from this study will contribute to the understanding of biomechanical events, under mechanical tension and torque, involving extension-torsion coupling of filaments.     

Geometrical and Mechanical Properties Control Actin Filament Organization

The different actin structures governing eukaryotic cell shape and movement are not only determined by the properties of the actin filaments and associated proteins, but also by geometrical constraints. We recently demonstrated that limiting nucleation to specific regions was sufficient to obtain actin networks with different organization. To further investigate how spatially constrained actin nucleation determines the emergent actin organization, we performed detailed simulations of the actin filament system using Cytosim. We first calibrated the steric interaction between filaments, by matching, in simulations and experiments, the bundled actin organization observed with a rectangular bar of nucleating factor. We then studied the overall organization of actin filaments generated by more complex pattern geometries used experimentally. We found that the fraction of parallel versus antiparallel bundles is determined by the mechanical properties of actin filament or bundles and the efficiency of nucleation. Thus nucleation geometry, actin filaments local interactions, bundle rigidity, and nucleation efficiency are the key parameters controlling the emergent actin architecture. We finally simulated more complex nucleation patterns and performed the corresponding experiments to confirm the predictive capabilities of the model.     

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.     

A mechanochemical model of actin filaments

In eukaryotic cells, actin filaments are involved in important processes such as motility, division, cell shape regulation, contractility, and mechanosensation. Actin filaments are polymerized chains of monomers, which themselves undergo a range of chemical events such as ATP hydrolysis, polymerization, and depolymerization. When forces are applied to F-actin, in addition to filament mechanical deformations, the applied force must also influence chemical events in the filament. We develop an intermediate-scale model of actin filaments that combines actin chemistry with filament-level deformations. The model is able to compute mechanical responses of F-actin during bending and stretching. The model also describes the interplay between ATP hydrolysis and filament deformations, including possible force-induced chemical state changes of actin monomers in the filament. The model can also be used to model the action of several actin-associated proteins, and for large-scale simulation of F-actin networks. All together, our model shows that mechanics and chemistry must be considered together to understand cytoskeletal dynamics in living cells.     

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.     

Effect of tensile force on the mechanical behavior of actin filaments

Actin filaments are the most abundant components of the cellular cytoskeleton, and play critical roles in various cellular functions such as migration, division and shape control. In these activities, mechanical tension causes structural changes in the double-helical structure of the actin filament, which is a key modulator of cytoskeletal reorganization. This study performed large-scale molecular dynamics (MD) and steered MD simulations to quantitatively analyze the effects of tensile force on the mechanical behavior of actin filaments. The results revealed that when a tensile force of 200pN was applied to a filament consisting of 14 actin subunits, the twist angle of the filament decreased by approximately 20°, corresponding to a rotation of approximately -2° per subunit, representing a critical structural change in actin filaments. Based on these structural changes, the variance in filament length and twist angle was found to decrease, leading to increases in extensional and torsional stiffness. Torsional stiffness increased significantly under the tensile condition, and the ratio of filament stiffness under tensile force to that under no external force increased significantly on longer temporal scales. The results obtained from this study contribute to the understanding of mechano-chemical interactions concerning actin dynamics, showing that increased tensile force in the filament prevents actin regulatory proteins from binding to the filament.     

Strain field in actin filament network in lamellipodia of migrating cells: Implication for network reorganization

Cell motility is spatiotemporally regulated by interactions among mechanical and biochemical factors involved in the regulation of cytoskeletal actin structure reorganization. Although the molecular mechanisms underlying cell motility have been well investigated, the contributions of mechanical factors such as strain in the network reorganization remain unclear. In this study, we have quantitatively evaluated the strain field in the actin filament network forming the lamellipodia of migrating fish keratocytes to elucidate the mechanism by which actin filament network reorganization is regulated by biomechanical factors. The results highlight the existence of a negative (compressive) strain in the lamellipodia whose direction is parallel to that of cell movement. A close correlation was found between the distributions of the strain and the actin filament density in the lamellipodia, suggesting that negative strain may be involved in filament depolymerization. Based on this result, we propose a selective depolymerization model which suggests that negative strain may couple with biomechanical factors such as ADF/cofilin to promote selective depolymerization of filaments oriented in the direction of the deformation because such filaments experience relatively higher levels of the deformation. This model, in conjunction with others, may explain the observed reduction in filament density and the reorganization of actin filament network at the back of the lamellipodia of migrating fish keratocytes. Thus, we suggest that by coupling with biochemical factors, mechanical factors are involved in the regulation of actin filament depolymerization, thereby contributing to the regulation of cell motility.     

Actin filaments function as a tension sensor by tension-dependent binding of cofilin to the filament

Intracellular and extracellular mechanical forces affect the structure and dynamics of the actin cytoskeleton. However, the underlying molecular and biophysical mechanisms, including how mechanical forces are sensed, are largely unknown. Actin-depolymerizing factor/cofilin proteins are actin-modulating proteins that are ubiquitously distributed in eukaryotes, and they are the most likely candidate as proteins to drive stress fiber disassembly in response to changes in tension in the fiber. In this study, we propose a novel hypothesis that tension in an actin filament prevents the filament from being severed by cofilin. To test this, we placed single actin filaments under tension using optical tweezers. When a fiber was tensed, it was severed after the application of cofilin with a significantly larger delay in comparison with control filaments suspended in solution. The binding rate of cofilin to an actin bundle decreased when the bundle was tensed. These results suggest that tension in an actin filament reduces the cofilin binding, resulting in a decrease in its effective severing activity.     

Computational Tension Mapping of Adherent Cells Based on Actin Imaging

Forces transiting through the cytoskeleton are known to play a role in adherent cell activity. Up to now few approaches haves been able to determine theses intracellular forces. We thus developed a computational mechanical model based on a reconstruction of the cytoskeleton of an adherent cell from fluorescence staining of the actin network and focal adhesions (FA). Our custom made algorithm converted the 2D image of an actin network into a map of contractile interactions inside a 2D node grid, each node representing a group of pixels. We assumed that actin filaments observed under fluorescence microscopy, appear brighter when thicker, we thus presumed that nodes corresponding to pixels with higher actin density were linked by stiffer interactions. This enabled us to create a system of heterogeneous interactions which represent the spatial organization of the contractile actin network. The contractility of this interaction system was then adapted to match the level of force the cell truly exerted on focal adhesions; forces on focal adhesions were estimated from their vinculin expressed size. This enabled the model to compute consistent mechanical forces transiting throughout the cell. After computation, we applied a graphical approach on the original actin image, which enabled us to calculate tension forces throughout the cell, or in a particular region or even in single stress fibers. It also enabled us to study different scenarios which may indicate the mechanical role of other cytoskeletal components such as microtubules. For instance, our results stated that the ratio between intra and extra cellular compression is inversely proportional to intracellular tension.     

The organization and regulation of the macrophage actin skeleton

To move, leukocytes extend portions of their cortical cytoplasm as pseudopods. These pseudopods are filled with a three-dimensional actin filament skeleton, the reversible assembly of which in response to receptor stimulation is thought to play a major role in providing the mechanical force for these protrusive movements. The organization of this actin skeleton occurs at different levels within the cell, and a number of macrophage proteins have been isolated and shown to affect the architecture, assembly, stability, and length of actin filaments in vitro. The architecture of cytoplasmic actin is regulated by proteins that cross-link filaments in higher-order structures. Actin-binding protein plays a major role in defining network structure by cross-linking actin filaments into orthogonal networks. Gelsolin may have a central role in regulating network structure. It binds to the sides of actin filaments and severs them, and binds the "barbed" filament end, thereby blocking monomer addition at this end. Gelsolin is activated to bind actin filaments by microM calcium. Dissociation of gelsolin bound on filament ends occurs in the presence of the polyphosphoinositides, PIP and PIP2. Calcium and PIP2 have been shown to be intracellular messengers of cell stimulation.     

Form-Finding Model Shows How Cytoskeleton Network Stiffness Is Realized

In eukaryotic cells the actin-cytoskeletal network provides stiffness and the driving force that contributes to changes in cell shape and cell motility, but the elastic behavior of this network is not well understood. In this paper a two dimensional form-finding model is proposed to investigate the elasticity of the actin filament network. Utilizing an initially random array of actin filaments and actin-cross-linking proteins the form-finding model iterates until the random array is brought into a stable equilibrium configuration. With some care given to actin filament density and length, distance between host sites for cross-linkers, and overall domain size the resulting configurations from the form-finding model are found to be topologically similar to cytoskeletal networks in real cells. The resulting network may then be mechanically exercised to explore how the actin filaments deform and align under load and the sensitivity of the network’s stiffness to actin filament density, length, etc. Results of the model are consistent with the experimental literature, e.g. actin filaments tend to re-orient in the direction of stretching; and the filament relative density, filament length, and actin-cross-linking protein’s relative density, control the actin-network stiffness. The model provides a ready means of extension to more complicated domains and a three-dimensional form-finding model is under development as well as models studying the formation of actin bundles.     

Micromechanics and ultrastructure of actin filament networks crosslinked by human fascin: a comparison with alpha-actinin.

Fascin is an actin crosslinking protein that organizes actin filaments into tightly packed bundles believed to mediate the formation of cellular protrusions and to provide mechanical support to stress fibers. Using quantitative rheological methods, we studied the evolution of the mechanical behavior of filamentous actin (F-actin) networks assembled in the presence of human fascin. The mechanical properties of F-actin/fascin networks were directly compared with those formed by alpha-actinin, a prototypical actin filament crosslinking/bundling protein. Gelation of F-actin networks in the presence of fascin (fascin to actin molar ratio >1:50) exhibits a non-monotonic behavior characterized by a burst of elasticity followed by a slow decline over time. Moreover, the rate of gelation shows a non-monotonic dependence on fascin concentration. In contrast, alpha-actinin increased the F-actin network elasticity and the rate of gelation monotonically. Time-resolved multiple-angle light scattering and confocal and electron microscopies suggest that this unique behavior is due to competition between fascin-mediated crosslinking and side-branching of actin filaments and bundles, on the one hand, and delayed actin assembly and enhanced network micro-heterogeneity, on the other hand. The behavior of F-actin/fascin solutions under oscillatory shear of different frequencies, which mimics the cell's response to forces applied at different rates, supports a key role for fascin-mediated F-actin side-branching. F-actin side-branching promotes the formation of interconnected networks, which completely inhibits the motion of actin filaments and bundles. Our results therefore show that despite sharing seemingly similar F-actin crosslinking/bundling activity, alpha-actinin and fascin display completely different mechanical behavior. When viewed in the context of recent microrheological measurements in living cells, these results provide the basis for understanding the synergy between multiple crosslinking proteins, and in particular the complementary mechanical roles of fascin and alpha-actinin in vivo.     

Apparent stiffness of vimentin intermediate filaments in living cells and its relation with other cytoskeletal polymers

The cytoskeleton is a complex network of interconnected biopolymers intimately involved in the generation and transmission of forces. Several mechanical properties of microtubules and actin filaments have been extensively explored in cells. In contrast, intermediate filaments (IFs) received comparatively less attention despite their central role in defining cell shape, motility and adhesion during physiological processes as well as in tumor progression. Here, we explored relevant biophysical properties of vimentin IFs in living cells combining confocal microscopy and a filament tracking routine that allows localizing filaments with ~20 nm precision. A Fourier-based analysis showed that IFs curvatures followed a thermal-like behavior characterized by an apparent persistence length (lp*) similar to that measured in aqueous solution. Additionally, we determined that certain perturbations of the cytoskeleton affect lp* and the lateral mobility of IFs as assessed in cells in which either the microtubule dynamic instability was reduced or actin filaments were partially depolymerized. Our results provide relevant clues on how vimentin IFs mechanically couple with microtubules and actin filaments in cells and support a role of this network in the response to mechanical stress.     

Interplay between passive tension and strong and weak binding cross- bridges in insect indirect flight muscle. A functional dissection by gelsolin-mediated thin filament removal

The interplay between passive and active mechanical properties of indirect flight muscle of the waterbug (Lethocerus) was investigated. A functional dissection of the relative contribution of cross-bridges, actin filaments, and C filaments to tension and stiffness of passive, activated, and rigor fibers was carried out by comparing mechanical properties at different ionic strengths of sarcomeres with and without thin filaments. Selective thin filament removal was accomplished by treatment with the actin-severving protein gelsolin. Thin filament, removal had no effect on passive tension, indicating that the C filament and the actin filament are mechanically independent and that passive tension is developed by the C filament in response to sarcomere stretch. Passive tension increased steeply with sarcomere length until an elastic limit was reached at only 6-7% sarcomere extension, which corresponds to an extension of 350% of the C filament. The passive tension-length relation of insect flight muscle was analyzed using a segmental extension model of passive tension development (Wang, K, R. McCarter, J. Wright, B. Jennate, and R Ramirez-Mitchell. 1991. Proc. Natl. Acad. Sci. USA. 88:7101-7109). Thin filament removal greatly depressed high frequency passive stiffness (2.2 kHz) and eliminated the ionic strength sensitivity of passive stiffness. It is likely that the passive stiffness component that is removed by gelsolin is derived from weak-binding cross-bridges, while the component that remains is derived from the C filament. Our results indicate that a significant number of weak-binding cross-bridges exist in passive insect muscle at room temperature and at an ionic strength of 195 mM. Analysis of rigor muscle indicated that while rigor tension is entirely actin based, rigor stiffness contains a component that resists gelsolin treatment and is therefore likely to be C filament based. Active tension and active stiffness of unextracted fibers were directly proportional to passive tension before activation. Similarly, passive stiffness due to weak bridges also increased linearly with passive tension, up to a limit. These correlations lead us to propose a stress-activation model for insect flight muscle in which passive tension is a prerequisite for the formation of both weak-binding and strong-binding cross-bridges.     

Action of cytochalasin D on cytoskeletal networks

Extraction of SC-1 cells (African green monkey kidney) with the detergent Triton X-100 in combination with stereo high-voltage electron microscopy of whole mount preparations has been used as an approach to determine the mode of action of cytochalasin D on cells. The cytoskeleton of extracted BSC-1 cells consists of substrate-associated filament bundles (stress fibers) and a highly cross-linked network of four major filament types extending throughout the cell body; 10-nm filaments, actin microfilaments, microtubules, and 2- to 3-nm filaments. Actin filaments and 2- to 3-nm filaments form numerous end- to-side contacts with other cytoskeletal filaments. Cytochalasin D treatment severely disrupts network organization, increases the number of actin filament ends, and leads to the formation of filamentous aggregates or foci composed mainly of actin filaments. Metabolic inhibitors prevent filament redistribution, foci formation, and cell arborization, but not disorganization of the three-dimensional filament network. In cells first extracted and then treated with cytochalasin D, network organization is disrupted, and the number of free filament ends is increased. Supernates of preparations treated in this way contain both short actin filaments and network fragments (i.e., actin filaments in end-to-side contact with other actin filaments). It is proposed that the dramatic effects of cytochalasin D on cells result from both a direct interaction of the drug with the actin filament component of cytoskeletal networks and a secondary cellular response. The former leads to an immediate disruption of the ordered cytoskeletal network that appears to involve breaking of actin filaments, rather than inhibition of actin filament-filament interactions (i.e., disruption of end-to-side contacts). The latter engages network fragments in an energy-dependent (contractile) event that leads to the formation of filament foci.     

Binding of myotrophin/V-1 to actin-capping protein: implications for how capping protein binds to the filament barbed end

The heterodimeric actin-capping protein (CP) regulates actin assembly and cell motility by binding tightly to the barbed end of the actin filament. Here we demonstrate that myotrophin/V-1 binds directly to CP in a 1:1 molar ratio with a Kd of 10-50 nm. V-1 binding inhibited the ability of CP to cap the barbed ends of actin filaments. The actin-binding COOH-terminal region, the "tentacle," of the CP beta subunit was important for binding V-1, with lesser contributions from the alpha subunit COOH-terminal region and the body of the protein. V-1 appears to be unable to bind to CP that is on the barbed end, based on the observations that V-1 had no activity in an uncapping assay and that the V-1.CP complex had no capping activity. Two loops of V-1, which extend out from the alpha-helical backbone of this ankyrin repeat protein, were necessary for V-1 to bind CP. Parallel computational studies determined a bound conformation of the beta tentacle with V-1 that is consistent with these findings, and they offered insight into experimentally observed differences between the alpha1 and alpha2 isoforms as well as the mutant lacking the alpha tentacle. These results support and extend our "wobble" model for CP binding to the actin filament, in which the two COOH-terminal regions of CP bind independently to the actin filament, and bound CP is able to wobble when attached only via its mobile beta-subunit tentacle. This model is also supported by molecular dynamics simulations of CP reported here. The existence of the wobble state may be important for actin dynamics in cells.     

Origin of Twist-Bend Coupling in Actin Filaments

Actin filaments are semiflexible polymers that display large-scale conformational twisting and bending motions. Modulation of filament bending and twisting dynamics has been linked to regulatory actin-binding protein function, filament assembly and fragmentation, and overall cell motility. The relationship between actin filament bending and twisting dynamics has not been evaluated. The numerical and analytical experiments presented here reveal that actin filaments have a strong intrinsic twist-bend coupling that obligates the reciprocal interconversion of bending energy and twisting stress. We developed a mesoscopic model of actin filaments that captures key documented features, including the subunit dimensions, interaction energies, helicity, and geometrical constraints coming from the double-stranded structure. The filament bending and torsional rigidities predicted by the model are comparable to experimental values, demonstrating the capacity of the model to assess the mechanical properties of actin filaments, including the coupling between twisting and bending motions. The predicted actin filament twist-bend coupling is strong, with a persistence length of 0.15-0.4 μm depending on the actin-bound nucleotide. Twist-bend coupling is an emergent property that introduces local asymmetry to actin filaments and contributes to their overall elasticity. Up to 60% of the filament subunit elastic free energy originates from twist-bend coupling, with the largest contributions resulting under relatively small deformations. A comparison of filaments with different architectures indicates that twist-bend coupling in actin filaments originates from their double protofilament and helical structure.