Contractile stress generation by actomyosin gels

Molecular profiling of sponge deflation reveals an ancient relaxant-inflammatory response

A hallmark of animals is the coordination of whole-body movement. Neurons and muscles are central to this, yet coordinated movements also exist in sponges that lack these cell types. Sponges are sessile animals with a complex canal system for filter-feeding. They undergo whole-body movements resembling "contractions" that lead to canal closure and water expulsion. Here, we combine 3D optical coherence microscopy, pharmacology, and functional proteomics to elucidate anatomy, molecular physiology, and control of these movements. We find them driven by the relaxation of actomyosin stress fibers in epithelial canal cells, which leads to whole-body deflation via collapse of the incurrent and expansion of the excurrent system, controlled by an Akt/NO/PKG/A pathway. A concomitant increase in reactive oxygen species and secretion of proteinases and cytokines indicate an inflammation-like state reminiscent of vascular endothelial cells experiencing oscillatory shear stress. This suggests an ancient relaxant-inflammatory response of perturbed fluid-carrying systems in animals.

Generic stress rectification in nonlinear elastic media

Stress propagation in nonlinear media is crucial in cell biology, where molecular motors exert anisotropic force dipoles on the fibrous cytoskeleton. While the force dipoles can be either contractile or expansile, a medium made of fibers which buckle under compression rectifies these stresses towards a biologically crucial contraction. A general understanding of this rectification phenomenon as a function of the medium's elasticity is however lacking. Here we use theoretical continuum elasticity to show that rectification is actually a very general effect in nonlinear materials subjected to anisotropic internal stresses. We analytically show that both bucklable and constitutively linear materials subjected to geometrical nonlinearities rectify small forces towards contraction, while granular-like materials rectify towards expansion. Using simulations, we moreover show that these results extend to larger forces. Beyond fiber networks, these results could shed light on the propagation of stresses in brittle or granular materials following a local plastic rearrangement.

Protein friction and filament bending facilitate contraction of disordered actomyosin networks

We use mathematical modeling and computation to investigate how protein friction facilitates contraction of disordered actomyosin networks. We simulate two-dimensional networks using an agent-based model, consisting of a system of force-balance equations for myosin motor proteins and semiflexible actin filaments. A major advantage of our approach is that it enables direct calculation of the network stress tensor, which provides a quantitative measure of contractility. Exploiting this, we use repeated simulations of disordered networks to confirm that both protein friction and actin filament bending are required for contraction. We then use simulations of elementary two-filament systems to show that filament bending flexibility can facilitate contraction on the microscopic scale. Finally, we show that actin filament turnover is necessary to sustain contraction and prevent filament aggregation. Simulations with and without turnover also exhibit contractile pulses. However, these pulses are aperiodic, suggesting that periodic pulsation can only arise because of additional regulatory mechanisms or more complex mechanical behavior.

Network Contractility During Cytokinesis-from Molecular to Global Views

Cytokinesis is the last stage of cell division, which partitions the mother cell into two daughter cells. It requires the assembly and constriction of a contractile ring that consists of a filamentous contractile network of actin and myosin. Network contractility depends on network architecture, level of connectivity and myosin motor activity, but how exactly is the contractile ring network organized or interconnected and how much it depends on motor activity remains unclear. Moreover, the contractile ring is not an isolated entity; rather, it is integrated into the surrounding cortex. Therefore, the mechanical properties of the cell cortex and cortical behaviors are expected to impact contractile ring functioning. Due to the complexity of the process, experimental approaches have been coupled to theoretical modeling in order to advance its global understanding. While earlier coarse-grained descriptions attempted to provide an integrated view of the process, recent models have mostly focused on understanding the behavior of an isolated contractile ring. Here we provide an overview of the organization and dynamics of the actomyosin network during cytokinesis and discuss existing theoretical models in light of cortical behaviors and experimental evidence from several systems. Our view on what is missing in current models and should be tested in the future is provided.

Strain energy storage and dissipation rate in active cell mechanics

When living cells are observed at rest on a flat substrate, they can typically exhibit a rounded (symmetric) or an elongated (polarized) shape. Although the cells are apparently at rest, the active stress generated by the molecular motors continuously stretches and drifts the actin network, the cytoskeleton of the cell. In this paper we theoretically compare the energy stored and dissipated in this active system in two geometric configurations of interest: symmetric and polarized. We find that the stored energy is larger for a radially symmetric cell at low activation regime, while the polar configuration has larger strain energy when the active stress is beyond a critical threshold. Conversely, the dissipation of energy in a symmetric cell is always larger than that of a nonsymmetric one. By a combination of symmetry arguments and competition between surface and bulk stress, we argue that radial symmetry is an energetically expensive metastable state that provides access to an infinite number of lower-energy states, the polarized configurations.

Force percolation of contractile active gels

Living systems provide a paradigmatic example of active soft matter. Cells and tissues comprise viscoelastic materials that exert forces and can actively change shape. This strikingly autonomous behavior is powered by the cytoskeleton, an active gel of semiflexible filaments, crosslinks, and molecular motors inside cells. Although individual motors are only a few nm in size and exert minute forces of a few pN, cells spatially integrate the activity of an ensemble of motors to produce larger contractile forces (∼nN and greater) on cellular, tissue, and organismal length scales. Here we review experimental and theoretical studies on contractile active gels composed of actin filaments and myosin motors. Unlike other active soft matter systems, which tend to form ordered patterns, actin-myosin systems exhibit a generic tendency to contract. Experimental studies of reconstituted actin-myosin model systems have long suggested that a mechanical interplay between motor activity and the network's connectivity governs this contractile behavior. Recent theoretical models indicate that this interplay can be understood in terms of percolation models, extended to include effects of motor activity on the network connectivity. Based on concepts from percolation theory, we propose a state diagram that unites a large body of experimental observations. This framework provides valuable insights into the mechanisms that drive cellular shape changes and also provides design principles for synthetic active materials.

Antagonistic Behaviors of NMY-1 and NMY-2 Maintain Ring Channels in the C. elegans Gonad

Contractile rings play critical roles in a number of biological processes, including oogenesis, wound healing, and cytokinesis. In many cases, the activity of motor proteins such as nonmuscle myosins is required for appropriate constriction of these contractile rings. In the gonad of the nematode worm Caenorhabditis elegans, ring channels are a specialized form of contractile ring that are maintained at a constant diameter before oogenesis. We propose a model of ring channel maintenance that explicitly incorporates force generation by motor proteins that can act normally or tangentially to the ring channel opening. We find that both modes of force generation are needed to maintain the ring channels. We demonstrate experimentally that the type II myosins NMY-1 and NMY-2 antagonize each other in the ring channels by producing force in perpendicular directions: the experimental depletion of NMY-1/theoretical decrease in orthogonal force allows premature ring constriction and cellularization, whereas the experimental depletion of NMY-2/theoretical decrease in tangential force opens the ring channels and prevents cellularization. Together, our experimental and theoretical results show that both forces, mediated by NMY-1 and NMY-2, are crucial for maintaining the appropriate ring channel diameter and dynamics throughout the gonad.

A Combination of Actin Treadmilling and Cross-Linking Drives Contraction of Random Actomyosin Arrays

We investigate computationally the self-organization and contraction of an initially random actomyosin ring. In the framework of a detailed physical model for a ring of cross-linked actin filaments and myosin-II clusters, we derive the force balance equations and solve them numerically. We find that to contract, actin filaments have to treadmill and to be sufficiently cross linked, and myosin has to be processive. The simulations reveal how contraction scales with mechanochemical parameters. For example, they show that the ring made of longer filaments generates greater force but contracts slower. The model predicts that the ring contracts with a constant rate proportional to the initial ring radius if either myosin is released from the ring during contraction and actin filaments shorten, or if myosin is retained in the ring, while the actin filament number decreases. We demonstrate that a balance of actin nucleation and compression-dependent disassembly can also sustain contraction. Finally, the model demonstrates that with time pattern formation takes place in the ring, worsening the contractile process. The more random the actin dynamics are, the higher the contractility will be.

MEDYAN: Mechanochemical Simulations of Contraction and Polarity Alignment in Actomyosin Networks

Active matter systems, and in particular the cell cytoskeleton, exhibit complex mechanochemical dynamics that are still not well understood. While prior computational models of cytoskeletal dynamics have lead to many conceptual insights, an important niche still needs to be filled with a high-resolution structural modeling framework, which includes a minimally-complete set of cytoskeletal chemistries, stochastically treats reaction and diffusion processes in three spatial dimensions, accurately and efficiently describes mechanical deformations of the filamentous network under stresses generated by molecular motors, and deeply couples mechanics and chemistry at high spatial resolution. To address this need, we propose a novel reactive coarse-grained force field, as well as a publicly available software package, named the Mechanochemical Dynamics of Active Networks (MEDYAN), for simulating active network evolution and dynamics (available at www.medyan.org). This model can be used to study the non-linear, far from equilibrium processes in active matter systems, in particular, comprised of interacting semi-flexible polymers embedded in a solution with complex reaction-diffusion processes. In this work, we applied MEDYAN to investigate a contractile actomyosin network consisting of actin filaments, alpha-actinin cross-linking proteins, and non-muscle myosin IIA mini-filaments. We found that these systems undergo a switch-like transition in simulations from a random network to ordered, bundled structures when cross-linker concentration is increased above a threshold value, inducing contraction driven by myosin II mini-filaments. Our simulations also show how myosin II mini-filaments, in tandem with cross-linkers, can produce a range of actin filament polarity distributions and alignment, which is crucially dependent on the rate of actin filament turnover and the actin filament’s resulting super-diffusive behavior in the actomyosin-cross-linker system. We discuss the biological implications of these findings for the arc formation in lamellipodium-to-lamellum architectural remodeling. Lastly, our simulations produce force-dependent accumulation of myosin II, which is thought to be responsible for their mechanosensation ability, also spontaneously generating myosin II concentration gradients in the solution phase of the simulation volume.

Active matter systems have the distinct ability to convert energy from their surroundings into mechanical work, which gives rise to them having highly dynamic properties. Modeling active matter systems and capturing their complex behavior has been a great challenge in past years due to the many coupled interactions between their constituent parts, including not only distinct chemical and mechanical properties, but also feedback between them. One of the most intriguing biological active matter systems is the cell cytoskeleton, which can dynamically respond to chemical and mechanical cues to control cell structure and shape, playing a central role in many higher-order cellular processes. To model these systems and reproduce their behavior, we present a new modeling approach which combines the chemical, mechanical, and molecular transport aspects of active matter systems, all represented with equivalent complexity, while also allowing for various forms of mechanochemical feedback. This modeling approach, named MEDYAN, and software implementation is flexible so that a wide range of active matter systems can be simulated with a high level of detail, and ultimately can help to describe active matter phenomena, and in particular, the dynamics of the cell cytoskeleton. In this work, we have used MEDYAN to simulate a cytoskeletal network consisting of actin filaments, cross-linking proteins, and myosin II molecular motors. We found that these systems show rich dynamical behaviors, undergoing alignment and bundling transitions, with an emergent contractility, as the concentrations of myosin II and cross-linking proteins, as well as actin filament turnover rates, are varied.

Activity induces traveling waves, vortices and spatiotemporal chaos in a model actomyosin layer

Inspired by the actomyosin cortex in biological cells, we investigate the spatiotemporal dynamics of a model describing a contractile active polar fluid sandwiched between two external media. The external media impose frictional forces at the interface with the active fluid. The fluid is driven by a spatially-homogeneous activity measuring the strength of the active stress that is generated by processes consuming a chemical fuel. We observe that as the activity is increased over two orders of magnitude the active polar fluid first shows spontaneous flow transition followed by transition to oscillatory dynamics with traveling waves and traveling vortices in the flow field. In the flow-tumbling regime, the active polar fluid also shows transition to spatiotemporal chaos at sufficiently large activities. These results demonstrate that level of activity alone can be used to tune the operating point of actomyosin layers with qualitatively different spatiotemporal dynamics.

Constriction model of actomyosin ring for cytokinesis by fission yeast using a two-state sliding filament mechanism

We developed a model describing the structure and contractile mechanism of the actomyosin ring in fission yeast, Schizosaccharomyces pombe. The proposed ring includes actin, myosin, and α-actinin, and is organized into a structure similar to that of muscle sarcomeres. This structure justifies the use of the sliding-filament mechanism developed by Huxley and Hill, but it is probably less organized relative to that of muscle sarcomeres. Ring contraction tension was generated via the same fundamental mechanism used to generate muscle tension, but some physicochemical parameters were adjusted to be consistent with the proposed ring structure. Simulations allowed an estimate of ring constriction tension that reproduced the observed ring constriction velocity using a physiologically possible, self-consistent set of parameters. Proposed molecular-level properties responsible for the thousand-fold slower constriction velocity of the ring relative to that of muscle sarcomeres include fewer myosin molecules involved, a less organized contractile configuration, a low α-actinin concentration, and a high resistance membrane tension. Ring constriction velocity is demonstrated as an exponential function of time despite a near linear appearance. We proposed a hypothesis to explain why excess myosin heads inhibit constriction velocity rather than enhance it. The model revealed how myosin concentration and elastic resistance tension are balanced during cytokinesis in S. pombe.

Modeling large-scale dynamic processes in the cell: polarization, waves, and division

The past decade has witnessed significant developments in molecular biology techniques, fluorescent labeling, and super-resolution microscopy, and together these advances have vastly increased our quantitative understanding of the cell. This detailed knowledge has concomitantly opened the door for biophysical modeling on a cellular scale. There have been comprehensive models produced describing many processes such as motility, transport, gene regulation, and chemotaxis. However, in this review we focus on a specific set of phenomena, namely cell polarization, F-actin waves, and cytokinesis. In each case, we compare and contrast various published models, highlight the relevant aspects of the biology, and provide a sense of the direction in which the field is moving.

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.

Connecting local active forces to macroscopic stress in elastic media

In contrast with ordinary materials, living matter drives its own motion by generating active, out-of-equilibrium internal stresses. These stresses typically originate from localized active elements embedded in an elastic medium, such as molecular motors inside the cell or contractile cells in a tissue. While many large-scale phenomenological theories of such active media have been developed, a systematic understanding of the emergence of stress from the local force-generating elements is lacking. In this paper, we present a rigorous theoretical framework to study this relationship. We show that the medium's macroscopic active stress tensor is equal to the active elements' force dipole tensor per unit volume in both continuum and discrete linear homogeneous media of arbitrary geometries. This relationship is conserved on average in the presence of disorder, but can be violated in nonlinear elastic media. Such effects can lead to either a reinforcement or an attenuation of the active stresses, giving us a glimpse of the ways in which nature might harness microscopic forces to create active materials.

Dynamic network morphology and tension buildup in a 3D model of cytokinetic ring assembly

During fission yeast cytokinesis, actin filaments nucleated by cortical formin Cdc12 are captured by myosin motors bound to a band of cortical nodes and bundled by cross-linking proteins. The myosin motors exert forces on the actin filaments, resulting in a net pulling of the nodes into a contractile ring, while cross-linking interactions help align actin filaments and nodes into a single bundle. We used these mechanisms in a three-dimensional computational model of contractile ring assembly, with semiflexible actin filaments growing from formins at cortical nodes, capturing of filaments by neighboring nodes, and cross-linking among filaments through attractive interactions. The model was used to predict profiles of actin filament density at the cell cortex, morphologies of condensing node-filament networks, and regimes of cortical tension by varying the node pulling force and strength of cross-linking among actin filaments. Results show that cross-linking interactions can lead to confinement of actin filaments at the simulated cortical boundary. We show that the ring-formation region in parameter space lies close to regions leading to clumps, meshworks or double rings, and stars/cables. Since boundaries between regions are not sharp, transient structures that resemble clumps, stars, and meshworks can appear in the process of ring assembly. These results are consistent with prior experiments with mutations in actin-filament turnover regulators, myosin motor activity, and changes in the concentration of cross-linkers that alter the morphology of the condensing network. Transient star shapes appear in some simulations, and these morphologies offer an explanation for star structures observed in prior experimental images. Finally, we quantify tension along actin filaments and forces on nodes during ring assembly and show that the mechanisms describing ring assembly can also drive ring constriction once the ring is formed.

Actomyosin regulation and symmetry breaking in a model of polarization in the early Caenorhabditis elegans embryo : symmetry breaking in cell polarization

Polarization, whereby a cell defines a spatial axis by segregating specific determinants to distinct regions, is an essential and highly conserved biological process. The process of polarization is initiated by a cue that breaks an initially symmetric distribution of determinants, allowing for a spatially asymmetric redistribution. The nature of this cue is currently not well understood. Utilizing the conservation of polarization process and its determinants, we theoretically investigate the nature of the cue and the regulation of contractility that enables the establishment of polarity in early embryos of the nematode worm Caenorhabditis elegans (C. elegans). Our biologically based model, which consists of coupled partial differential equations, suggests that a biochemical but not mechanical cue is sufficient for symmetry breaking, and inhibition of contractile elements by specific determinants is needed for sustained spatial redistribution.

Early-time dynamics of actomyosin polarization in cells of confined shape in elastic matrices

The cell shape and the rigidity of the extracellular matrix have been shown to play an important role in the regulation of cytoskeleton structure and force generation. Elastic stresses that develop by actomyosin contraction feedback on myosin activity and govern the anisotropic polarization of stress fibers in the cell. We theoretically study the consequences that the cell shape and matrix rigidity may have on the dynamics and steady state polarization of actomyosin forces in the cell. Actomyosin forces are assumed to polarize in accordance with the stresses that develop in the cytoskeleton. The theory examines this self-polarization process as a relaxation response determined by two distinct susceptibility factors and two characteristic times. These reveal two canonical polarization responses to local variations in the elastic stress: an isotropic response, in which actomyosin dipolar stress isotropically changes in magnitude, and an orientational response, in which actomyosin forces orient with no net change in magnitude. Actual polarization may show up as a superimposition of the two mechanisms yielding different phases in the polarization response as observed experimentally. The cell shape and elastic moduli of the surroundings are shown to govern both the dynamics of the process as well as the steady-state. We predict that in the steady-state, beyond a critical matrix rigidity, spherical cells exert maximal force, and below that rigidity, elongated or flattened cells exert more force. Similar behaviors are reflected in the rate of the polarization process. The theory is also applicable to study the elastic response of whole cell aggregates in a gel.

Force to divide: structural and mechanical requirements for actomyosin ring contraction

One of the unresolved questions in the field of cell division is how the actomyosin cytoskeleton remains structurally organized while generating the contractile force to divide one cell into two. In analogy to the actomyosin-based force production mechanism in striated muscle, it was originally proposed that contractile stress in the actomyosin ring is generated via a sliding filament mechanism within an organized sarcomere-like array. However, over the last 30 years, ultrastructural and functional studies have noted important distinctions between cytokinetic structures in dividing cells and muscle sarcomeres. Myosin-II motor activity is not always required, and there is evidence that actin depolymerization contributes to contraction. In this Review, the architecture and contractile dynamics of the actomyosin ring at the cell division plane will be discussed. We will report the interdisciplinary advances in the field as well as their integration into a mechanistic understanding of contraction in cell division and in other biological processes that rely on an actomyosin-based force-generating system.

Stress generation by myosin minifilaments in actin bundles

Forces and stresses generated by the action of myosin minifilaments are analyzed in idealized computer-generated actin bundles, and compared to results for isotropic actin networks. The bundles are generated as random collections of actin filaments in two dimensions with constrained orientations, crosslinked and attached to two fixed walls. Myosin minifilaments are placed on actin filament pairs and allowed to move and deform the network so that it exerts forces on the walls. The vast majority of simulation runs end with contractile minifilament stress, because minifilaments rotate into energetically stable contractile configurations. This process is aided by the bending and stretching of actin filaments, which accomodate minifilament rotation. Stresses for bundles are greater than those for isotropic networks, and antiparallel filaments generate more tension than parallel filaments. The forces transmitted by the actin network to the walls of the simulation cell often exceed the tension in the minifilament itself.

Cortical geometry may influence placement of interface between Par protein domains in early Caenorhabditis elegans embryos

During polarization, proteins and other polarity determinants segregate to the opposite ends of the cell (the poles) creating biochemically and dynamically distinct regions. Embryos of the nematode worm Caenorhabditis elegans (C. elegans) polarize shortly after fertilization, creating distinct regions of Par protein family members. These regions are maintained through to first cleavage when the embryo divides along the plane specified by the interface between regions, creating daughter cells with different protein content. In wild type single cell embryos the interface between these Par protein regions is reliably positioned at approximately 60% egg length, however, it is not known what mechanisms are responsible for specifying the position of the interface. In this investigation, we use two mathematical models to investigate the movement and positioning of the interface: a biologically based reaction-diffusion model of Par protein dynamics, and the analytically tractable perturbed Allen-Cahn equation. When we numerically simulate the models on a static 2D domain with constant thickness, both models exhibit a persistently moving interface that specifies the boundary between distinct regions. When we modify the simulation domain geometry, movement halts and the interface is stably positioned where the domain thickness increases. Using asymptotic analysis with the perturbed Allen-Cahn equation, we show that interface movement depends explicitly on domain geometry. Using a combination of analytic and numeric techniques, we demonstrate that domain geometry, a historically overlooked aspect of cellular simulations, may play a significant role in spatial protein patterning during polarization.

Actin depolymerization drives actomyosin ring contraction during budding yeast cytokinesis

Actin filaments and myosin II are evolutionarily conserved force-generating components of the contractile ring during cytokinesis. Here we show that in budding yeast, actin filament depolymerization plays a major role in actomyosin ring constriction. Cofilin mutation or chemically stabilizing actin filaments attenuate actomyosin ring constriction. Deletion of myosin II motor domain or the myosin regulatory light chain reduced the contraction rate and also the rate of actin depolymerization in the ring. We constructed a quantitative microscopic model of actomyosin ring constriction via filament sliding driven by both actin depolymerization and myosin II motor activity. Model simulations based on experimental measurements support the notion that actin depolymerization is the predominant mechanism for ring constriction. The model predicts invariability of total contraction time regardless of the initial ring size, as originally reported for C. elegans embryonic cells. This prediction was validated in yeast cells of different sizes due to different ploidies.

Cytokinesis in animal cells

Cytokinesis, the final step in cell division, partitions the contents of a single cell into two. In animal cells, cytokinesis occurs through cortical remodeling orchestrated by the anaphase spindle. Cytokinesis relies on a tight interplay between signaling and cellular mechanics and has attracted the attention of both biologists and physicists for more than a century. In this review, we provide an overview of four topics in animal cell cytokinesis: (a) signaling between the anaphase spindle and cortex, (b) the mechanics of cortical remodeling, (c) abscission, and (d) regulation of cytokinesis by the cell cycle machinery. We report on recent progress in these areas and highlight some of the outstanding questions that these findings bring into focus.

Active contractility in actomyosin networks

Contractile forces are essential for many developmental processes involving cell shape change and tissue deformation. Recent experiments on reconstituted actomyosin networks, the major component of the contractile machinery, have shown that active contractility occurs above a threshold motor concentration and within a window of cross-link concentration. We present a microscopic dynamic model that incorporates two essential aspects of actomyosin self-organization: the asymmetric load response of individual actin filaments and the correlated motor-driven events mimicking myosin-induced filament sliding. Using computer simulations, we examine how the concentration and susceptibility of motors contribute to their collective behavior and interplay with the network connectivity to regulate macroscopic contractility. Our model is shown to capture the formation and dynamics of contractile structures and agree with the observed dependence of active contractility on microscopic parameters, including the contractility onset. Cooperative action of load-resisting motors in a force-percolating structure integrates local contraction/buckling events into a global contractile state via an active coarsening process, in contrast to the flow transition driven by uncorrelated kicks of susceptible motors.

Separation anxiety: stress, tension and cytokinesis

Cytokinesis, the physical separation of a mother cell into two daughter cells, progresses through a series of well-defined changes in morphology. These changes involve distinct biochemical and mechanical processes. Here, we review the mechanical features of cells during cytokinesis, discussing both the material properties as well as sources of stresses, both active and passive, which lead to the observed changes in morphology. We also describe a mechanosensory feedback control system that regulates protein localization and shape progression during cytokinesis.

Mechanotransduction is enhanced by the synergistic action of heterotypic cell interactions and TGF-β1

With the use of planar substrates and collagen gels, the field of mechanotransduction has focused on the role of extracellular matrix stiffness, mechanical tension, and TGF-β1 in generating a more contractile fibroblast. However, little is known about the role of cell-cell interactions in inducing cellular contraction. We used 3-dimensional self-assembled microtissues, in which cell-cell interactions dominate, and a recently developed cell power assay (an assay for mechanotransduction) to quantify the effects of TGF-β1 vs. the heterotypic cell interface on the power exerted by pure normal human fibroblast (NHF) and pure rat hepatocyte 35 (H35) microtissues and their mixes. As a control, we found that TGF-β1 only doubled the power output of pure NHF and pure H35 microtissues, whereas the heterotypic environment resulted in a 5-fold increase in cell power (0.24±0.05 to 1.17±0.13 fJ/h). Seeding TGF-β1-treated NHFs with untreated H35 cells demonstrated that the heterotypic environment and TGF-β1 synergistically increase cell power by 22× by maximizing heterotypic cell interactions. Using a mathematical simulation of stress generation, we showed that tensile forces can be enhanced by heterotypic cell interactions. These data render a new understanding of how heterotypic cell interactions may increase cellular force generation during wound healing.

Modeling morphodynamic phenotypes and dynamic regimes of cell motion

Many cellular processes and signaling pathways converge onto cell morphology and cell motion, which share important components. The mechanisms used for propulsion could also be responsible for shape changes, if they are capable of generating the rich observed variety of dynamic regimes. Additionally, the analysis of cell shape changes in space and time promises insight into the state of the cytoskeleton and signaling pathways controlling it. While this has been obvious for some time by now, little effort has been made to systematically and quantitatively explore this source of information. First pioneering experimental work revealed morphodynamic phenotypes which can be associated with dynamic regimes like oscillations and excitability. Here, we review the current state of modeling of morphodynamic phenotypes, the experimental results and discuss the ideas on the mechanisms driving shape changes which are suggested by modeling.

Growing up is stressful: biophysical laws of morphogenesis

Would it not be nice to understand the rules that govern how a small and round zygote reforms itself into a full blown three-dimensional and structured organism? The past decades have provided us with a wealth of knowledge about molecular mechanisms, intracellular behaviors, and tissue organization. However, we still do not know how to systematically integrate molecular mechanisms into descriptions that operate at larger scales involving higher-order structures such as the actomyosin cell cortex or an entire tissue. For development, it is the biophysical laws by which these structures deform, move, and restructure that are essential for morphogenetic rearrangements at developmental length- and time-scales. Recent years have seen the advent of systematic approaches for identifying these laws and ways to determine associated physical behaviors. Here I attempt to paint an intuitive picture of the mechanical concepts that are important for large-scale developmental rearrangements, and I briefly review the technique of laser ablation for measuring associated physical quantities and testing physical models.

The emergence of sarcomeric, graded-polarity and spindle-like patterns in bundles of short cytoskeletal polymers and two opposite molecular motors

We use linear stability analysis and numerical solutions of partial differential equations to investigate pattern formation in the one-dimensional system of short dynamic polymers and one (plus-end directed) or two (one is plus-end, another minus-end directed) molecular motors. If polymer sliding and motor gliding rates are slow and/or the polymer turnover rate is fast, then the polymer-motor bundle has mixed polarity and homogeneous motor distribution. However, if motor gliding is fast, a sarcomeric pattern with periodic bands of alternating polymer polarity separated by motor aggregates evolves. On the other hand, if polymer sliding is fast, a graded-polarity bundle with motors at the center emerges. In the presence of the second, minus-end directed motor, the sarcomeric pattern is more ubiquitous, while the graded-polarity pattern is destabilized. However, if the minus-end motor is weaker than the plus-end directed one, and/or polymer nucleation is autocatalytic, and/or long polymers are present in the bundle, then a spindle-like architecture with a sorted-out polarity emerges with the plus-end motors at the center and minus-end motors at the edges. We discuss modeling implications for actin-myosin fibers and in vitro and meiotic spindles.

General mechanism of actomyosin contractility

Stress generation by myosin minifilaments is analyzed via simulation of their motion in a random actin network. The stresses are overwhelmingly contractile because minifilament equilibrium positions having contractile stress have lower energy than those for expansive stress. Force chains lead to unexpectedly large stresses.

Optimal matrix rigidity for stress fiber polarization in stem cells

The shape and differentiation of human mesenchymal stem cells is especially sensitive to the rigidity of their environment; the physical mechanisms involved are unknown. A theoretical model and experiments demonstrate here that the polarization/alignment of stress-fibers within stem cells is a non-monotonic function of matrix rigidity. We treat the cell as an active elastic inclusion in a surrounding matrix whose polarizability, unlike dead matter, depends on the feedback of cellular forces that develop in response to matrix stresses. The theory correctly predicts the monotonic increase of the cellular forces with the matrix rigidity and the alignment of stress-fibers parallel to the long axis of cells. We show that the anisotropy of this alignment depends non-monotonically on matrix rigidity and demonstrate it experimentally by quantifying the orientational distribution of stress-fibers in stem cells. These findings offer a first physical insight for the dependence of stem cell differentiation on tissue elasticity.

Cell shape, spreading symmetry and the polarization of stress-fibers in cells

The active regulation of cellular forces during cell adhesion plays an important role in the determination of cell size, shape and internal structure. While on flat, homogeneous and isotropic substrates some cells spread isotropically, others spread anisotropically and assume elongated structures. In addition, in their native environment as well as in vitro experiments, the cell shape and spreading asymmetry can be modulated by the local distribution of adhesive molecules and topography of the environment. We present a simple elastic model, and experiments on stem cells to explain the variation of cell size with the matrix rigidity. In addition, we predict the experimental consequences of two mechanisms of acto-myosin polarization and focus here on the effect of the cell spreading asymmetry on the regulation of the stress-fiber alignment in the cytoskeleton. We show that when cell spreading is sufficiently asymmetric the alignment of acto-myosin forces in the cell increases monotonically with the matrix rigidity; however, in general this alignment is non-monotonic as shown previously. These results highlight the importance of the symmetry characteristics of cell spreading in the regulation of cytoskeleton structure and suggest a mechanism by which different cell types may acquire different morphologies and internal structures in different mechanical environments.

Cytoskeletal dynamics in fission yeast: a review of models for polarization and division

We review modeling studies concerning cytoskeletal activity of fission yeast. Recent models vary in length and time scales, describing a range of phenomena from cellular morphogenesis to polymer assembly. The components of cytoskeleton act in concert to mediate cell-scale events and interactions such as polarization. The mathematical models reduce these events and interactions to their essential ingredients, describing the cytoskeleton by its bulk properties. On a smaller scale, models describe cytoskeletal subcomponents and how bulk properties emerge.

Theoretical concepts and models of cellular mechanosensing

Recent discoveries have established that mechanical properties of the cellular environment such as its rigidity, geometry, and external stresses play an important role in determining the cellular function and fate. Mechanical properties have been shown to influence cell shape and orientation, regulate cell proliferation and differentiation, and even govern the development and organization of tissues. In recent years, many theoretical and experimental investigations have been carried out to elucidate the mechanisms and consequences of the mechanosensitivity of cells. In this review, we discuss recent theoretical concepts and approaches that explain and predict cell mechanosensitivity. We focus on the interplay of active and passive processes that govern cell-cell and cell-matrix interactions and discuss the role of this interplay in the processes of cell adhesion, regulation of cytoskeleton mechanics and the response of cells to applied mechanical stresses.

Mechanics of the F-actin cytoskeleton

Dynamic regulation of the filamentous actin (F-actin) cytoskeleton is critical to numerous physical cellular processes, including cell adhesion, migration and division. Each of these processes require precise regulation of cell shape and mechanical force generation which, to a large degree, is regulated by the dynamic mechanical behaviors of a diverse assortment of F-actin networks and bundles. In this review, we review the current understanding of the mechanics of F-actin networks and identify areas of further research needed to establish physical models. We first review our understanding of the mechanical behaviors of F-actin networks reconstituted in vitro, with a focus on the nonlinear mechanical response and behavior of "active" F-actin networks. We then explore the types of mechanical response measured of cytoskeletal F-actin networks and bundles formed in living cells and identify how these measurements correspond to those performed on reconstituted F-actin networks formed in vitro. Together, these approaches identify the challenges and opportunities in the study of living cytoskeletal matter.

Motor-induced sliding of microtubule and actin bundles

Interactions of multiple molecular motors with bundles of actin and microtubule filaments form the basis for many cytoskeletal processes including axonal growth, muscle contraction, cell division and platelet formation. Continuum models based on generalized diffusion equations have been suggested to quantify the dynamics of such active bundles. In highly cross-linked and densely packed filament bundles, however, a major complication arises due to the multiple interactions that each filament forms with its neighbors. To explore the effects of these interactions, we used detailed computer simulations and studied the bundles with different types of motors at different densities and boundary conditions. We found that highly cross-linked bundles exhibit effects of long-ranged interactions that are sensitive to the boundary conditions. In open bundles, these give rise to 'telescopic' patterns resulting in significant acceleration of the filaments at the edges. In contrast, in ringed bundles, the long-ranged interactions 'lock' filaments and slow down their movements. The filaments in loosely connected bundles, on the other hand, undergo local diffusion-drift dynamics consistent with previous continuum models. Our simulations also demonstrate the sorting phenomena in the mixed-polarity bundles and reveal characteristic scales and conditions for spontaneous pattern formation in the bundle. We discuss the relevance of our results for cytoskeleton systems such as microtubules in axons, platelet formation, kinetochore fibers and actin bundles in motile cells.

Mathematical models and simulations of cellular processes based on actin filaments

Actin filaments help to maintain the physical integrity of cells and participate in many processes that produce cellular movements. Studies of the processes that depend on actin filaments have progressed to the point where mathematical models and computer simulations are an essential part of the experimental toolkit. These quantitative models integrate knowledge about the structures of the key proteins and the rate and equilibrium constants for the reactions for comparison with a growing body of quantitative measurements of dynamic processes in live cells. Models and simulations are essential because it is impossible to appreciate by intuition alone the properties that emerge from a network of coupled reactions, particularly when the system contains many components, and force is one of the parameters.

A quantitative analysis of contractility in active cytoskeletal protein networks

Cells actively produce contractile forces for a variety of processes including cytokinesis and motility. Contractility is known to rely on myosin II motors which convert chemical energy from ATP hydrolysis into forces on actin filaments. However, the basic physical principles of cell contractility remain poorly understood. We reconstitute contractility in a simplified model system of purified F-actin, muscle myosin II motors, and alpha-actinin cross-linkers. We show that contractility occurs above a threshold motor concentration and within a window of cross-linker concentrations. We also quantify the pore size of the bundled networks and find contractility to occur at a critical distance between the bundles. We propose a simple mechanism of contraction based on myosin filaments pulling neighboring bundles together into an aggregated structure. Observations of this reconstituted system in both bulk and low-dimensional geometries show that the contracting gels pull on and deform their surface with a contractile force of approximately 1 microN, or approximately 100 pN per F-actin bundle. Cytoplasmic extracts contracting in identical environments show a similar behavior and dependence on myosin as the reconstituted system. Our results suggest that cellular contractility can be sensitively regulated by tuning the (local) activity of molecular motors and the cross-linker density and binding affinity.

Model of polarization and bistability of cell fragments

Directed cell motility is preceded by cell polarization-development of a front-rear asymmetry of the cytoskeleton and the cell shape. Extensive studies implicated complex spatial-temporal feedbacks between multiple signaling pathways in establishing cell polarity, yet physical mechanisms of this phenomenon remain elusive. Based on observations of lamellipodial fragments of fish keratocyte cells, we suggest a purely thermodynamic (not involving signaling) quantitative model of the cell polarization and bistability. The model is based on the interplay between pushing force exerted by F-actin polymerization on the cell edges, contractile force powered by myosin II across the cell, and elastic tension in the cell membrane. We calculate the thermodynamic work produced by these intracellular forces, and show that on the short timescale, the cell mechanics can be characterized by an effective energy profile with two minima that describe two stable states separated by an energy barrier and corresponding to the nonpolarized and polarized cells. Cell dynamics implied by this energy profile is bistable-the cell is either disk-shaped and stationary, or crescent-shaped and motile-with a possible transition between them upon a finite external stimulus able to drive the system over the macroscopic energy barrier. The model accounts for the observations of the keratocyte fragments' behavior and generates quantitative predictions about relations between the intracellular forces' magnitudes and the cell geometry and motility.