A master relation defines the nonlinear viscoelasticity of single fibroblasts

Cell mechanical functions such as locomotion, contraction, and division are controlled by the cytoskeleton, a dynamic biopolymer network whose mechanical properties remain poorly understood. We perform single-cell uniaxial stretching experiments on 3T3 fibroblasts. By superimposing small amplitude oscillations on a mechanically prestressed cell, we find a transition from linear viscoelastic behavior to power law stress stiffening. Data from different cells over several stress decades can be uniquely scaled to obtain a master relation between the viscoelastic moduli and the average force. Remarkably, this relation holds independently of deformation history, adhesion biochemistry, and intensity of active contraction. In particular, it is irrelevant whether force is actively generated by the cell or externally imposed by stretching. We propose that the master relation reflects the mechanical behavior of the force-bearing actin cytoskeleton, in agreement with stress stiffening known from semiflexible filament networks.     

A microstructurally informed model for the mechanical response of three-dimensional actin networks

We propose a class of microstructurally informed models for the linear elastic mechanical behaviour of cross-linked polymer networks such as the actin cytoskeleton. Salient features of the models include the possibility to represent anisotropic mechanical behaviour resulting from anisotropic filament distributions, and a power law scaling of the mechanical properties with the filament density. Mechanical models within the class are parameterized by seven different constants. We demonstrate a procedure for determining these constants using finite element models of three-dimensional actin networks. Actin filaments and cross-links were modelled as elastic rods, and the networks were constructed at physiological volume fractions and at the scale of an image voxel. We show the performance of the model in estimating the mechanical behaviour of the networks over a wide range of filament densities and degrees of anisotropy.     

A microstructurally informed model for the mechanical response of three-dimensional actin networks

We propose a class of microstructurally informed models for the linear elastic mechanical behaviour of cross-linked polymer networks such as the actin cytoskeleton. Salient features of the models include the possibility to represent anisotropic mechanical behaviour resulting from anisotropic filament distributions, and a power law scaling of the mechanical properties with the filament density. Mechanical models within the class are parameterized by seven different constants. We demonstrate a procedure for determining these constants using finite element models of three-dimensional actin networks. Actin filaments and cross-links were modelled as elastic rods, and the networks were constructed at physiological volume fractions and at the scale of an image voxel. We show the performance of the model in estimating the mechanical behaviour of the networks over a wide range of filament densities and degrees of anisotropy.     

Mechanics and multiple-particle tracking microheterogeneity of alpha-actinin-cross-linked actin filament networks

Cell morphology is controlled by the actin cytoskeleton organization and mechanical properties, which are regulated by the available contents in actin and actin regulatory proteins. Using rheometry and the recently developed multiple-particle tracking method, we compare the mechanical properties and microheterogeneity of actin filament networks containing the F-actin cross-linking protein alpha-actinin. The elasticity of F-actin/alpha-actinin networks increases with actin concentration more rapidly for a fixed molar ratio of actin to alpha-actinin than in the absence of alpha-actinin, for networks of fixed alpha-actinin concentration and of fixed actin concentration, but more slowly than theoretically predicted for a homogeneous cross-linked semiflexible polymer network. These rheological measurements are complemented by multiple-particle tracking of fluorescent microspheres imbedded in the networks. The distribution of the mean squared displacements of these microspheres becomes progressively more asymmetric and wider for increasing concentration in alpha-actinin and, to a lesser extent, for increasing actin concentration, which suggests that F-actin networks become progressively heterogeneous for increasing protein content. This may explain the slower-than-predicted rise in elasticity of F-actin/alpha-actinin networks. Together these in vitro results suggest that actin and alpha-actinin provides the cell with an unsuspected range of regulatory pathways to modulate its cytoskeleton's micromechanics and local organization in vivo.     

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.     

Substrate deformation determines actin cytoskeleton reorganization: A mathematical modeling and experimental study

A mathematical model has been developed to define the relationship between the actin cytoskeleton reorganization of a cell and substrate deformation acting on the cell. The model is based on the following major assumptions: (a) normal substrate strain, not the shear substrate strain, determines the actin cytoskeleton reorganization; (b) the normal substrate strain is transmitted to individual actin filaments; (c) each actin filament has a basal strain energy (BSE) when the cell adheres to the substrate without stretching; and (d) the actin filaments undergo disassembly when their strain energies are decreased to zero or increased to twice their BSEs. The resulting model predicts that the actin filaments are formed in the direction where their BSEs are minimally altered. This direction is therefore the one without normal substrate strain. The prediction was confirmed by experiments conducted on both fibroblasts and endothelial cells. The present model may be relevant for understanding better the effects of mechanical stimuli on the cells.     

Cytoskeletal bundle mechanics

The mechanical properties of cytoskeletal actin bundles play an essential role in numerous physiological processes, including hearing, fertilization, cell migration, and growth. Cells employ a multitude of actin-binding proteins to actively regulate bundle dimensions and cross-linking properties to suit biological function. The mechanical properties of actin bundles vary by orders of magnitude depending on diameter and length, cross-linking protein type and concentration, and constituent filament properties. Despite their importance to cell function, the molecular design principles responsible for this mechanical behavior remain unknown. Here, we examine the mechanics of cytoskeletal bundles using a molecular-based model that accounts for the discrete nature of constituent actin filaments and their distinct cross-linking proteins. A generic competition between filament stretching and cross-link shearing determines three markedly different regimes of mechanical response that are delineated by the relative values of two simple design parameters, revealing the universal nature of bundle-bending mechanics. In each regime, bundle-bending stiffness displays distinct scaling behavior with respect to bundle dimensions and molecular composition, as observed in reconstituted actin bundles in vitro. This mechanical behavior has direct implications on the physiological bending, buckling, and entropic stretching behavior of cytoskeletal processes, as well as reconstituted actin systems. Results are used to predict the bending regimes of various in vivo cytoskeletal bundles that are not easily accessible to experiment and to generate hypotheses regarding implications of the isolated behavior on in vivo bundle function.     

Microheterogeneity controls the rate of gelation of actin filament networks

Rapid sol-gel transitions of the actin cytoskeleton are required for many key cellular processes, including cell spreading and cell locomotion. Actin monomers assemble into semiflexible polymers that rapidly intertwine into a network, a process that in vitro takes approximately 1 min for an actin concentration of 1 mg/ml. The same actin filament network, however, takes approximately 1 h to exhibit a steady-state elasticity. We hypothesize that the slow gelation of F-actin is due to the slow establishment of a homogeneous meshwork. Using a novel method, time-resolved multiple particle tracking, which monitors the range of thermally excited displacements of microspheres imbedded in the network, we show that the increase in elasticity in a polymerizing solution of actin parallels the progressive decline of the network microheterogeneity. The rates of gelation and network homogenization slightly decrease with actin concentration and in the presence of the F-actin cross-linking proteins alpha-actinin and fascin, whereas the rate of actin polymerization increases dramatically with actin concentration. Our measurements show that the slow spatial homogenization of the actin filament network, not actin polymerization or the formation of polymer overlaps, is the rate-limiting step in the establishment of an elastic actin network and suggest that a new activity of F-actin binding proteins may be required for the rapid formation of a homogeneous stiff gel.     

Emergence and maintenance of variable-length actin filaments in a limiting pool of building blocks

Actin is one of the key structural components of the eukaryotic cytoskeleton that regulates cellular architecture and mechanical properties. Dynamic regulation of actin filament length and organization is essential for the control of many physiological processes including cell adhesion, motility and division. While previous studies have mostly focused on the mechanisms controlling the length of single actin filaments, it remains poorly understood how distinct actin filament populations in cells maintain different lengths using the same set of molecular building blocks. Here, we develop a theoretical model for the length regulation of multiple actin filaments by nucleation and growth-rate modulation by actin-binding proteins in a limiting pool of monomers. We first show that spontaneous nucleation of actin filaments naturally leads to heterogeneities in filament length distribution. We then investigate the effects of filament growth inhibition by capping proteins and growth promotion by formin proteins on filament length distribution. We find that filament length heterogeneity can be increased by growth inhibition, whereas growth promoters do not significantly affect length heterogeneity. Interestingly, a competition between filament growth inhibitors and growth promoters can give rise to bimodal filament length distribution as well as a highly heterogeneous length distribution with large statistical dispersion. We quantitatively predict how heterogeneity in actin filament length can be modulated by tuning filamentous actin nucleation and growth rates in order to create distinct filament subpopulations with different lengths.     

Multiple-particle tracking measurements of heterogeneities in solutions of actin filaments and actin bundles

One of the central functions of actin cytoskeleton is to provide the mechanical support required for the establishment and maintenance of cell morphology. The mechanical properties of actin filament assemblies are a consequence of both the available polymer concentration and the actin regulatory proteins that direct the formation of higher order structures. By monitoring the displacement of well-dispersed microspheres via fluorescence microscopy, we probe the degree of spatial heterogeneity of F-actin gels and networks in vitro. We compare the distribution of the time-dependent mean-square displacement (MSD) of polystyrene microspheres imbedded in low- and high-concentration F-actin solutions, in the presence and absence of the F-actin-bundling protein fascin. The MSD distribution of a 2. 6-microM F-actin solution is symmetric and its standard deviation is similar to that of a homogeneous solution of glycerol of similar zero-shear viscosity. However, increasing actin concentration renders the MSD distribution wide and asymmetric, an effect enhanced by fascin. Quantitative changes in the shape of the MSD distribution correlate qualitatively with the presence of large heterogeneities in F-actin solutions produced by increased filament concentration and the presence of actin bundles, as detected by confocal microscopy. Multiple-particle tracking offers a new, quantitative method to characterize the organization of biopolymers in solution.     

Three-dimensional reconstruction of the actin cytoskeleton from stereo images

In eucaryotic cells, actin filaments are abundant components in the cytoskeleton where they form a complex three dimensional (3D) structural network that provides the cell with its shape and mechanical properties. However, understanding the structural and mechanical properties of actin filaments composing the cell cytoskeleton is often hampered by the inability to faithfully reconstruct the three-dimensional geometric relationships. This paper presents a vision-based reconstruction approach that automatically reconstitutes the three-dimensional structures of cytoskeletal polymers from stereo image pairs taken at the different tilt angles. The approach finds corresponding points between two images and recovers the depth information about the structures. The computational process consists of three major procedures: feature representation, stereo matching, and disparity refinement, implemented in a multi-resolution manner based on a coarse-to-fine strategy. The reconstruction depicts the three-dimensional structure of cytoskeletal polymers and their geometric relationships. New and useful information becomes available and allows quantitative analysis of the structure. Measurement of the cytoskeleton geometrical properties and the filament concentration in a defined volume are obtained by direct calculation.     

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.     

Modulation of actin mechanics by caldesmon and tropomyosin

The ability of cells to sense and respond to physiological forces relies on the actin cytoskeleton, a dynamic structure that can directly convert forces into biochemical signals. Because of the association of muscle actin-binding proteins (ABPs) may affect F-actin and hence cytoskeleton mechanics, we investigated the effects of several ABPs on the mechanical properties of the actin filaments. The structural interactions between ABPs and helical actin filaments can vary between interstrand interactions that bridge azimuthally adjacent actin monomers between filament strands (i.e. by molecular stapling as proposed for caldesmon) or, intrastrand interactions that reinforce axially adjacent actin monomers along strands (i.e. as in the interaction of tropomyosin with actin). Here, we analyzed thermally driven fluctuations in actin's shape to measure the flexural rigidity of actin filaments with different ABPs bound. We show that the binding of phalloidin increases the persistence length of actin by 1.9-fold. Similarly, the intrastrand reinforcement by smooth and skeletal muscle tropomyosins increases the persistence length 1.5- and 2- fold respectively. We also show that the interstrand crosslinking by the C-terminal actin-binding fragment of caldesmon, H32K, increases persistence length by 1.6-fold. While still remaining bound to actin, phosphorylation of H32K by ERK abolishes the molecular staple (Foster et al. 2004. J Biol Chem 279;53387-53394) and reduces filament rigidity to that of actin with no ABPs bound. Lastly, we show that the effect of binding both smooth muscle tropomyosin and H32K is not additive. The combination of structural and mechanical studies on ABP-actin interactions will help provide information about the biophysical mechanism of force transduction in cells.     

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.     

Building a dendritic actin filament network branch by branch: models of filament orientation pattern and force generation in lamellipodia

We review mathematical and computational models of the structure, dynamics, and force generation properties of dendritic actin networks. These models have been motivated by the dendritic nucleation model, which provided a mechanistic picture of how the actin cytoskeleton system powers cell motility. We describe how they aimed to explain the self-organization of the branched network into a bimodal distribution of filament orientations peaked at 35° and ??35° with respect to the direction of membrane protrusion, as well as other patterns. Concave and convex force–velocity relationships were derived, depending on network organization, filament, and membrane elasticity and accounting for actin polymerization at the barbed end as a Brownian ratchet. This review also describes models that considered the kinetics and transport of actin and diffuse regulators and mechanical coupling to a substrate, together with explicit modeling of dendritic networks.     

Building distinct actin filament networks in a common cytoplasm

Eukaryotic cells generate a diversity of actin filament networks in a common cytoplasm to optimally perform functions such as cell motility, cell adhesion, endocytosis and cytokinesis. Each of these networks maintains precise mechanical and dynamic properties by autonomously controlling the composition of its interacting proteins and spatial organization of its actin filaments. In this review, we discuss the chemical and physical mechanisms that target distinct sets of actin-binding proteins to distinct actin filament populations after nucleation, resulting in the assembly of actin filament networks that are optimized for specific functions.     

Simulations of dynamically cross-linked actin networks: Morphology,rheology, and hydrodynamic interactions

Cross-linked actin networks are the primary component of the cell cytoskeleton and have been the subject of numerous experimental and modeling studies. While these studies have demonstrated that the networks are viscoelastic materials, evolving from elastic solids on short timescales to viscous fluids on long ones, questions remain about the duration of each asymptotic regime, the role of the surrounding fluid, and the behavior of the networks on intermediate timescales. Here we perform detailed simulations of passively cross-linked non-Brownian actin networks to quantify the principal timescales involved in the elastoviscous behavior, study the role of nonlocal hydrodynamic interactions, and parameterize continuum models from discrete stochastic simulations. To do this, we extend our recent computational framework for semiflexible filament suspensions, which is based on nonlocal slender body theory, to actin networks with dynamic cross linkers and finite filament lifetime. We introduce a model where the cross linkers are elastic springs with sticky ends stochastically binding to and unbinding from the elastic filaments, which randomly turn over at a characteristic rate. We show that, depending on the parameters, the network evolves to a steady state morphology that is either an isotropic actin mesh or a mesh with embedded actin bundles. For different degrees of bundling, we numerically apply small-amplitude oscillatory shear deformation to extract three timescales from networks of hundreds of filaments and cross linkers. We analyze the dependence of these timescales, which range from the order of hundredths of a second to the actin turnover time of several seconds, on the dynamic nature of the links, solvent viscosity, and filament bending stiffness. We show that the network is mostly elastic on the short time scale, with the elasticity coming mainly from the cross links, and viscous on the long time scale, with the effective viscosity originating primarily from stretching and breaking of the cross links. We show that the influence of nonlocal hydrodynamic interactions depends on the network morphology: for homogeneous meshworks, nonlocal hydrodynamics gives only a small correction to the viscous behavior, but for bundled networks it both hinders the formation of bundles and significantly lowers the resistance to shear once bundles are formed. We use our results to construct three-timescale generalized Maxwell models of the networks.     

Organization and dynamics of cross-linked actin filaments in confined environments

The organization of the actin cytoskeleton is impacted by the interplay between physical confinement, features of cross-linking proteins, and deformations of semiflexible actin filaments. Some cross-linking proteins preferentially bind filaments in parallel, although others bind more indiscriminately. However, a quantitative understanding of how the mode of binding influences the assembly of actin networks in confined environments is lacking. Here we employ coarse-grained computer simulations to study the dynamics and organization of semiflexible actin filaments in confined regions upon the addition of cross-linkers. We characterize how the emergent behavior is influenced by the system shape, the number and type of cross-linking proteins, and the length of filaments. Structures include isolated clusters of filaments, highly connected filament bundles, and networks of interconnected bundles and loops. Elongation of one dimension of the system promotes the formation of long bundles that align with the elongated axis. Dynamics are governed by rapid cross-linking into aggregates, followed by a slower change in their shape and connectivity. Cross-linking decreases the average bending energy of short or sparsely connected filaments by suppressing shape fluctuations. However, it increases the average bending energy in highly connected networks because filament bundles become deformed, and small numbers of filaments exhibit long-lived, highly unfavorable configurations. Indiscriminate cross-linking promotes the formation of high-energy configurations due to the increased likelihood of unfavorable, difficult-to-relax configurations at early times. Taken together, this work demonstrates physical mechanisms by which cross-linker binding and physical confinement impact the emergent behavior of actin networks, which is relevant both in cells and in synthetic environments.     

Mechanisms of actin disassembly

The actin cytoskeleton is constantly assembling and disassembling. Cells harness the energy of these turnover dynamics to drive cell motility and organize cytoplasm. Although much is known about how cells control actin polymerization, we do not understand how actin filaments depolymerize inside cells. I briefly describe how the combination of imaging actin filament dynamics in cells and using in vitro biochemistry progressively altered our views of actin depolymerization. I describe why I do not think that the prevailing model of actin filament turnover—cofilin-mediated actin filament severing—can account for actin filament disassembly detected in cells. Finally, I speculate that cells might be able to tune the mechanism of actin depolymerization to meet physiological demands and selectively control the stabilities of different actin arrays.