A predictive model of cell traction forces based on cell geometry

Recent work has indicated that the shape and size of a cell can influence how a cell spreads, develops focal adhesions, and exerts forces on the substrate. However, it is unclear how cell shape regulates these events. Here we present a computational model that uses cell shape to predict the magnitude and direction of forces generated by cells. The predicted results are compared to experimentally measured traction forces, and show that the model can predict traction force direction, relative magnitude, and force distribution within the cell using only cell shape as an input. Analysis of the model shows that the magnitude and direction of the traction force at a given point is proportional to the first moment of area about that point in the cell, suggesting that contractile forces within the cell act on the entire cytoskeletal network as a single cohesive unit. Through this model, we demonstrate that intrinsic properties of cell shape can facilitate changes in traction force patterns, independently of heterogeneous mechanical properties or signaling events within the cell.     

Invited review: engineering approaches to cytoskeletal mechanics.

An outstanding problem in cell biology is how cells sense mechanical forces and how those forces affect cellular functions. Various biophysical and biochemical mechanisms have been invoked to answer this question. A growing body of evidence indicates that the deformable cytoskeleton (CSK), an intracellular network of interconnected filamentous biopolymers, provides a physical basis for transducing mechanical signals into biochemical signals. Therefore, to understand how mechanical forces regulate cellular functions, it is important to know how cells respond to changes in the CSK force balance and to identify the underlying mechanisms that control transmission of mechanical forces throughout the CSK and bring it to equilibrium. Recent developments of new experimental techniques for measuring cell mechanical properties and novel theoretical models of cellular mechanics make it now possible to identify and quantitate the contributions of various CSK structures to the overall balance of mechanical forces in the cell. This review focuses on engineering approaches that have been used in the past two decades in studies of the mechanics of the CSK.     

Structural microtubule cap: stability,catastrophe, rescue,and third state

Microtubules polymerize from GTP-liganded tubulin dimers, but are essentially made of GDP-liganded tubulin. We investigate the tug-of-war resulting from the fact that GDP-liganded tubulin favors a curved configuration, but is forced to remain in a straight one when part of a microtubule. We point out that near the end of a microtubule, the proximity of the end shifts the balance in this tug-of-war, with some protofilament bending as result. This somewhat relaxes the microtubule lattice near its end, resulting in a structural cap. This structural cap thus is a simple mechanical consequence of two well-established facts: protofilaments made of GDP-liganded tubulin have intrinsic curvature, and microtubules are elastic, made from material that can yield to forces, in casu its own intrinsic forces. We explore possible properties of this structural cap, and demonstrate 1) how it allows both polymerization from GTP-liganded tubulin and rapid depolymerization in its absence; 2) how rescue can occur; 3) how a third, meta-stable intermediate state is possible and can explain some experimental results; and 4) how the tapered tips observed at polymerizing microtubule ends are stabilized during growth, though unable to accommodate a lateral cap. This scenario thus supports the widely accepted GTP-cap model by suggesting a stabilizing mechanism that explains the many aspects of dynamic instability.     

Actin Mechanics and Fragmentation

Cell physiological processes require the regulation and coordination of both mechanical and dynamical properties of the actin cytoskeleton. Here we review recent advances in understanding the mechanical properties and stability of actin filaments and how these properties are manifested at larger (network) length scales. We discuss how forces can influence local biochemical interactions, resulting in the formation of mechanically sensitive dynamic steady states. Understanding the regulation of such force-activated chemistries and dynamic steady states reflects an important challenge for future work that will provide valuable insights as to how the actin cytoskeleton engenders mechanoresponsiveness of living cells.     

Variation and robustness of the mechanics of gastrulation: the role of tissue mechanical properties during morphogenesis

Diverse mechanisms of morphogenesis generate a wide variety of animal forms. In this work, we discuss two ways that the mechanical properties of embryonic tissues could guide one of the earliest morphogenetic movements in animals, gastrulation. First, morphogenetic movements are a function of both the forces generated by cells and the mechanical properties of the tissues. Second, cells could change their behavior in response to their mechanical environment. Theoretical studies of gastrulation indicate that different morphogenetic mechanisms differ in their inherent sensitivity to tissue mechanical properties. Those few empirical studies that have investigated the mechanical properties of amphibian and echinoderm gastrula-stage embryos indicate that there could be high embryo-to-embryo variability in tissue stiffness. Such high embryo-to-embryo variability would imply that gastrulation is fairly robust to variation in tissue stiffness. Cell culture studies demonstrate a wide variety of cellular responses to the mechanical properties of their microenvironment. These responses are likely to be developmentally regulated, and could either increase or decrease the robustness of gastrulation movements depending on which cells express which responses. Hence both passive physical and mechanoregulatory processes will determine how sensitive gastrulation is to tissue mechanics. Addressing these questions is important for understanding the significance of diverse programs of early development, and how genetic or environmental perturbations influence development. We discuss methods for measuring embryo-to-embryo variability in tissue mechanics, and for experimentally perturbing those mechanical properties to determine the sensitivity of gastrulation to tissue mechanics.     

Periodic Forces Trigger a Complex Mechanical Response in Ubiquitin

Mechanical forces govern physiological processes in all living organisms. Many cellular forces, for example, those generated in cyclic conformational changes of biological machines, have repetitive components. In apparent contrast, little is known about how dynamic protein structures respond to periodic mechanical information. Ubiquitin is a small protein found in all eukaryotes. We developed molecular dynamics simulations to unfold single and multimeric ubiquitins with periodic forces. By using a coarse-grained representation, we were able to model forces with periods about 2 orders of magnitude longer than the protein's relaxation time. We found that even a moderate periodic force weakened the protein and shifted its unfolding pathways in a frequency- and amplitude-dependent manner. A complex dynamic response with secondary structure refolding and an increasing importance of local interactions was revealed. Importantly, repetitive forces with broadly distributed frequencies elicited very similar molecular responses compared to fixed-frequency forces. When testing the influence of pulling geometry on ubiquitin's mechanical stability, it was found that the linkage involved in the mechanical degradation of cellular proteins renders the protein remarkably insensitive to periodic forces. We also devised a complementary kinetic energy landscape model that traces these observations and explains periodic-force, single-molecule measurements. In turn, this analytical model is capable of predicting dynamic protein responses. These results provide new insights into ubiquitin mechanics and a potential mechanical role during protein degradation, as well as first frameworks for dynamic protein stability and the modeling of repetitive mechanical processes.     

How Do Benthic Organisms Withstand Moving Water?

Many aquatic plants and animals spend part of their lives anchoredto the substratum as water flows by. There are a number of mechanismsby which such sessile organisms can affect the magnitude ofthe flow-induced forces they encounter, as well as the distributionand magnitude of the mechanical stresses in their bodies producedby those forces. Furthermore, the mechanical properties of theskeletal tissues of such organisms affect how much they deformand whether or not they will break in response to flow-inducedstresses. There are different mechanisms by which organismscan withstand the water flow characterizing a particular typeof habitat. Biomechanics is a useful tool for studying how theperformance of organisms depends on their structure. Biomechanicalstudies should be accompanied by knowledge of the natural historyand ecology of the organisms in question if they are to leadto insights about how organisms work.     

Developmental bias in evolution: evolutionary accessibility of phenotypes in a model evo-devo system

SUMMARY The success of the modern synthesis has resulted in forces of evolutionary change other than natural selection being marginalized. However, recent work has attempted to show the importance of non-selective influences in shaping organic form. One such force is developmental bias, in which phenotypes are differentially produced. We use a simulation model of neural development to explore questions of general interest about developmental systems. From this analysis, we find that the pattern of developmental bias varies strongly with the genotype even among phenotypically-neutral genotypes. In addition to this genotype-dependent developmental bias ( local bias ), an intrinsic bias exists in the developmental system ( global bias ). We also show that developmental bias varies among related genotypes that produce the same phenotype. Finally, we illustrate how a pattern of bias emerges from the manner in which mutations affect the regulatory structure of the wild-type genotype. These results suggest that developmental bias could have a strong influence on the direction of evolutionary modification.     

Decoupling Substrate Stiffness, Spread Area, and Micropost Density: A Close Spatial Relationship between Traction Forces and Focal Adhesions

Mechanical cues can influence the manner in which cells generate traction forces and form focal adhesions. The stiffness of a cell's substrate and the available area on which it can spread can influence its generation of traction forces, but to what extent these factors are intertwined is unclear. In this study, we used microcontact printing and micropost arrays to control cell spreading, substrate stiffness, and post density to assess their effect on traction forces and focal adhesions. We find that both the spread area and the substrate stiffness influence traction forces in an independent manner, but these factors have opposite effects: cells on stiffer substrates produce higher average forces, whereas cells with larger spread areas generate lower average forces. We show that post density influences the generation of traction forces in a manner that is more dominant than the effect of spread area. Additionally, we observe that focal adhesions respond to spread area, substrate stiffness, and post density in a manner that closely matches the trends seen for traction forces. This work supports the notion that traction forces and focal adhesions have a close relationship in their response to mechanical cues.     

Transmission of Mechanical Stresses within the Cytoskeleton of Adherent Cells: A Theoretical Analysis Based on a Multi-Component Cell Model

How environmental mechanical forces affect cellular functions is a central problem in cell biology. Theoretical models of cellular biomechanics provide relevant tools for understanding how the contributions of deformable intracellular components and specific adhesion conditions at the cell interface are integrated for determining the overall balance of mechanical forces within the cell. We investigate here the spatial distributions of intracellular stresses when adherent cells are probed by magnetic twisting cytometry. The influence of the cell nucleus stiffness on the simulated nonlinear torque-bead rotation response is analyzed by considering a finite element multi-component cell model in which the cell and its nucleus are considered as different hyperelastic materials. We additionally take into account the mechanical properties of the basal cell cortex, which can be affected by the interaction of the basal cell membrane with the extracellular substrate. In agreement with data obtained on epithelial cells, the simulated behaviour of the cell model relates the hyperelastic response observed at the entire cell scale to the distribution of stresses and strains within the nucleus and the cytoskeleton, up to cell adhesion areas. These results, which indicate how mechanical forces are transmitted at distant points through the cytoskeleton, are compared to recent data imaging the highly localized distribution of intracellular stresses.     

Hybridization and introgression in two ecologically dissimilar Fundulus hybrid zones

Hybridization and introgression appear more common in rapidly evolving groups, suggesting an important role in the evolutionary process. Detailed studies of how extrinsic or intrinsic forces regulate hybridization and introgression have the potential for broadening our understanding of mechanisms generating diversity. Species in the Fundulus notatus species complex have broad overlapping ranges and occur in replicated hybrid zones along predictable stream gradients. Typical hybrid zone structure has Fundulus olivaceus in headwaters, F. notatus downstream, and hybrid zones near confluences or abrupt shifts in habitat. Rarely, the typical upstream‐downstream orientation is reversed raising questions as to how hybrid zones are formed and maintained. We used next‐generation sequencing data to study hybridization and introgression in hybrid zones in neighboring drainages that differ in orientation (typical and reversed). We predicted extrinsic forces linked to stream gradients would result in noticeable differences between the two. Contrary to predictions, the data indicate the hybrid zones are remarkably similar. We used individual‐based simulations to explore the potential role of intrinsic and extrinsic forces in generating and maintaining typical and reversed hybrid zones. Simulation results were consistent with reversed hybrid zones being formed from stochastic processes combined with strong intrinsic forces and weak extrinsic forces.     

Combined Force-Torque Spectroscopy of Proteins by Means of Multiscale Molecular Simulation

Assessing the structural properties of large proteins is important to gain an understanding of their function in, e.g., biological systems or biomedical applications. We propose a method to examine the mechanical properties of proteins subject to applied forces by means of multiscale simulation. Both stretching and torsional forces are considered, and these may be applied independently of each other. As a proof of principle, we apply torsional forces to a coarse-grained continuum model of the antibody protein immunoglobulin G using fluctuating finite element analysis and use it to identify the area of strongest deformation. This region is essential to the torsional properties of the molecule as a whole because it represents the softest, most deformable domain. Zooming in, this part of the molecule is subjected to torques and stretching forces using molecular dynamics simulations on an atomistically resolved level to investigate its torsional properties. We calculate the torsional resistance as a function of the rotation of the domain while subjecting it to various stretching forces. From this, we assess how the measured twist-torque profiles develop with increasing stretching force and show that they exhibit torsion stiffening, in qualitative agreement with experimental findings. We argue that combining the twist-torque profiles for various stretching forces effectively results in a combined force-torque spectroscopy analysis, which may serve as a mechanical signature for a biological macromolecule.     

Dissecting the influence of cellular senescence on cell mechanics and extracellular matrix formation in vitro

Tissue formation and healing both require cell proliferation and migration, but also extracellular matrix production and tensioning. In addition to restricting proliferation of damaged cells, increasing evidence suggests that cellular senescence also has distinct modulatory effects during wound healing and fibrosis. Yet, a direct role of senescent cells during tissue formation beyond paracrine signaling remains unknown. We here report how individual modules of the senescence program differentially influence cell mechanics and ECM expression with relevance for tissue formation. We compared DNA damage-mediated and DNA damage-independent senescence which was achieved through over-expression of either p16^Ink4a or p21^Cip1 cyclin-dependent kinase inhibitors in primary human skin fibroblasts. Cellular senescence modulated focal adhesion size and composition. All senescent cells exhibited increased single cell forces which led to an increase in tissue stiffness and contraction in an in vitro 3D tissue formation model selectively for p16 and p21-overexpressing cells. The mechanical component was complemented by an altered expression profile of ECM-related genes including collagens, lysyl oxidases, and MMPs. We found that particularly the lack of collagen and lysyl oxidase expression in the case of DNA damage-mediated senescence foiled their intrinsic mechanical potential. These observations highlight the active mechanical role of cellular senescence during tissue formation as well as the need to synthesize a functional ECM network capable of transferring and storing cellular forces.     

Changes in surface topologies of chondrocytes subjected to mechanical forces: an AFM analysis

The cartilage is composed of chondrocytes embedded in a matrix of collagen fibrils interspersed within a network of proteoglycans and is constantly exposed to biomechanical forces during normal joint movement. Characterization of the surface morphology, cytoskeletal structure, adherance and elastic properties of these mechanosensitive cells are crucial in understanding the effects of mechanical forces around a cell and how a cell responds to changes in its physical environment. In this work, we employed the atomic force microscope (AFM) to image cultured chondrocytes before and after subjecting them to mechanical forces in the presence or absence of interleukin-1β to mimic inflammatory conditions. Nanoscale imaging and quantitative measurements from AFM data revealed that there are distinct changes in cell-surface topology and cytoskeleton arrangement in the cells following treatment with mechanical forces, IL-1β or both. Our findings for the first time demonstrate that cultured chondrocytes are amenable to high-resolution AFM imaging and dynamic tensile forces may help overcome the effect of inflammatory factors on chondrocyte response.     

Mechanical stimuli on C2C12 myoblasts affect myoblast differentiation, focal adhesion kinase phosphorylation and galectin-1 expression: a proteomic approach

Mechanical forces are crucial in the regulation of cell morphology and function. At the cellular level, these forces influence myoblast differentiation and fusion. In this study, we applied mechanical stimuli to embryonic muscle cells using magnetic microbeads, a method shown to apply stress to specific receptors on the cell surface. We showed that mechanical stimuli promote an increase in FAK (focal adhesion kinase) phosphorylation. In order to further shed light in the process of myoblast-induced differentiation by mechanical stimuli, we performed a proteomic analysis. Thirteen proteins were found to be affected by mechanical stimulation including galectin-1, annexin III and RhoGDI (Rho guanine-nucleotide-dissociation inhibitor). In this study, we demonstrate how the combination of this method of mechanical stimuli and proteomic analysis can be a powerful tool to detect proteins that are potentially interacting in biochemical pathways or complex cellular mechanisms during the process of myoblast differentiation. We determined an increase in expression and changes in cellular localization of galectin-1 in mechanically stimulated myoblasts. A potential involvement of galectin-1 in myoblast differentiation is presented.     

Use of biorobotic models of highly deformable fins for studying the mechanics and control of fin forces in fishes

Bony fish swim with a level of agility that is unmatched in human-developed systems. This is due, in part, to the ability of the fish to carefully control hydrodynamic forces through the active modulation of the fins' kinematics and mechanical properties. To better understand how fish produce and control forces, biorobotic models of the bluegill sunfish's (Lepomis macrochirus) caudal fin and pectoral fins were developed. The designs of these systems were based on detailed analyses of the anatomy, kinematics, and hydrodynamics of the biological fins. The fin models have been used to investigate how fin kinematics and the mechanical properties of the fin-rays influence propulsive forces and to explore kinematic patterns that were inspired by biological motions but that were not explicitly performed by the fish. Results from studies conducted with the fin models indicate that subtle changes to the kinematics and mechanical properties of fin rays can significantly impact the magnitude, direction, and time course of the 3D forces used for propulsion and maneuvers. The magnitude of the force tends to scale with the fin's stiffness, but the direction of the force is not invariant, and this causes disproportional changes in the magnitude of the thrust, lift, and lateral components of force. Results from these studies shed light on the multiple strategies that are available to the fish to modulate fin forces.