A quantitative model of cellular elasticity based on tensegrity

Modelling the rheology of living cell cytoplasm: poroviscoelasticity and fluid-to-solid transition

Eukaryotic cell rheology has important consequences for vital processes such as adhesion, migration, and differentiation. Experiments indicate that cell cytoplasm can exhibit both elastic and viscous characteristics in different regimes, while the transport of fluid (cytosol) through the cross-linked filamentous scaffold (cytoskeleton) is reminiscent of mass transfer by diffusion through a porous medium. To gain insights into this complex rheological behaviour, we construct a computational model for the cell cytoplasm as a poroviscoelastic material formulated on the principles of nonlinear continuum mechanics, where we model the cytoplasm as a porous viscoelastic scaffold with an embedded viscous fluid flowing between the pores to model the cytosol. Baseline simulations (neglecting the viscosity of the cytosol) indicate that the system exhibits seven different regimes across the parameter space spanned by the viscoelastic relaxation timescale of the cytoskeleton and the poroelastic diffusion timescale; these regimes agree qualitatively with experimental measurements. Furthermore, the theoretical model also allows us to elucidate the additional role of pore fluid viscosity, which enters the system as a distinct viscous timescale. We show that increasing this viscous timescale hinders the passage of the pore fluid (reducing the poroelastic diffusion) and makes the cytoplasm rheology increasingly incompressible, shifting the phase boundaries between the regimes.

Biomechanical characterization and modeling of human mesenchymal stem cells under compression

The application of microelectromechanical systems (MEMS) in biomedical devices has expanded vastly over the last few decades, with MEMS devices being developed to measure different characteristics of cells. The study of cell mechanics offers valuable understanding of cell viability and functionality. Cell biomechanics approaches also facilitate the characterization of important cell and tissue behaviors. In particular, understanding of the biological response of cells to their biomechanical environment would enhance the knowledge of how cellular responses correlate to tissue level characteristics and how some diseases, such as cancer, grow in the body. This study focuses on viscoelastic modeling of the behavior of a single suspended human mesenchymal stem cell (hMSC). Mechanical properties of hMSC cells are particularly important in tissue engineering and research for the treatment of cardiovascular diseases. We evaluated the elastic and viscoelastic properties of hMSC cells using a miniaturized custom-made BioMEMS device. Our results were compared to the elastic and viscoelastic properties measured by other methods such as atomic force microscopy (AFM) and micropipette aspiration. Different approaches were applied to model the experimentally obtained force data, including elastic and Standard Linear Solid (SLS) constitutive models, and the corresponding constants were derived. These values were compared to the ones in literature that were based on micropipette aspiration and AFM methods. We then utilized a tensegrity approach to model major parts of the internal structure of the cell and treat the cell as a network of viscoelastic microtubules and microfilaments, as opposed to a simple spherical blob. The results predicted from the tensegrity model were similar to the recorded experimental data.

Altered mechanical properties of actin fibers due to breast cancer invasion: parameter identification based on micropipette aspiration and multiscale tensegrity modeling

The biophysical properties of cells change with cancer invasion to fulfill their metastatic behavior. Cell softening induced by cancer is highly associated with alterations in cytoskeleton fibers. Changes in the mechanical properties of cytoskeletal fibers have not been quantified due to technical limitations. In this study, we used the micropipette aspiration technique to calculate and compare the viscoelastic properties of non-invasive and invasive breast cancer cells. We evaluated the mechanical properties of actin fibers and microtubules of two cancerous cell lines by using multiscale tensegrity modeling and an optimization method. Cancer invasion caused altered viscoelastic behavior of cells and the results of modeling showed changes in mechanical properties of major cytoskeleton fibers. The stiffness and viscosity constant of actin fibers in non-invasive cells were 1.28 and 2.27 times higher than those of the invasive cells, respectively. However, changes in mechanical properties of microtubules were minor. Immunofluorescent staining of fibers and their quantified distributions confirmed altered actin distribution among two cell lines, in contrast to microtubule distribution. This study highlights the function of cytoskeletal fibers in cancer progression, which could be of interest in designing therapeutic strategies to target cancer progress. Firstly, the viscoelastic behavior of non-invasive and invasive cells is examined with micropipette aspiration tests. A tensegrity model of cells is developed to mimic the viscoelastic behavior of cells, and tensegrity element stiffness is evaluated in an optimization procedure based on micropipette aspiration tests. Finally, by using immunofluorescent staining and confocal imaging, mechanical properties of actin filaments and microtubules of cancer cells are investigated during the course of metastasis.

Modelling Cell Origami via a Tensegrity Model of the Cytoskeleton in Adherent Cells

Cell origami has been widely used in the field of three-dimensional (3D) cell-populated microstructures due to their multiple advantages, including high biocompatibility, the lack of special requirements for substrate materials, and the lack of damage to cells. A 3D finite element method (FEM) model of an adherent cell based on the tensegrity structure is constructed to describe cell origami by using the principle of the origami folding technique and cell traction forces. Adherent cell models contain a cytoskeleton (CSK), which is primarily composed of microtubules (MTs), microfilaments (MFs), intermediate filaments (IFs), and a nucleoskeleton (NSK), which is mainly made up of the nuclear lamina and chromatin. The microplate is assumed to be an isotropic linear-elastic solid material with a flexible joint that is connected to the cell tensegrity structure model by spring elements representing focal adhesion complexes (FACs). To investigate the effects of the degree of complexity of the tensegrity structure and NSK on the folding angle of the microplate, four models are established in the study. The results demonstrate that the inclusion of the NSK can increase the folding angle of the microplate, indicating that the cell is closer to its physiological environment, while increased complexity can reduce the folding angle of the microplate since the folding angle is depended on the cell types. The proposed adherent cell FEM models are validated by comparisons with reported results. These findings can provide theoretical guidance for the application of biotechnology and the analysis of 3D structures of cells and have profound implications for the self-assembly of cell-based microscale medical devices.

A quantitative quantum-chemical analysis tool for the distribution of mechanical force in molecules

The promising field of mechanochemistry suffers from a general lack of understanding of the distribution and propagation of force in a stretched molecule, which limits its applicability up to the present day. In this article, we introduce the JEDI (Judgement of Energy DIstribution) analysis, which is the first quantum chemical method that provides a quantitative understanding of the distribution of mechanical stress energy among all degrees of freedom in a molecule. The method is carried out on the basis of static or dynamic calculations under the influence of an external force and makes use of a Hessian matrix in redundant internal coordinates (bond lengths, bond angles, and dihedral angles), so that all relevant degrees of freedom of a molecule are included and mechanochemical processes can be interpreted in a chemically intuitive way. The JEDI method is characterized by its modest computational effort, with the calculation of the Hessian being the rate-determining step, and delivers, except for the harmonic approximation, exact ab initio results. We apply the JEDI analysis to several example molecules in both static quantum chemical calculations and Born-Oppenheimer Molecular Dynamics simulations in which molecules are subject to an external force, thus studying not only the distribution and the propagation of strain in mechanically deformed systems, but also gaining valuable insights into the mechanochemically induced isomerization of trans-3,4-dimethylcyclobutene to trans,trans-2,4-hexadiene. The JEDI analysis can potentially be used in the discussion of sonochemical reactions, molecular motors, mechanophores, and photoswitches as well as in the development of molecular force probes.

Localized vibration of a microtubule surrounded by randomly distributed cross linkers

Based on finite element simulation, the present work studies free vibration of a microtubule surrounded by 3D randomly distributed cross linkers in living cells. A basic result of the present work is that transverse vibration modes associated with the lowest frequencies are highly localized, in sharp contrast to the through-length modes predicted by the commonly used classic elastic foundation model. Our simulations show that the deflected length of localized modes increases with increasing frequency and approaches the entire length of microtubule when frequency approaches the minimum classic frequency given by the elastic foundation model. In particular, unlike the length-sensitive classic frequencies predicted by the elastic foundation model, the lowest frequencies of localized modes predicted by the present model are insensitive to the length of microtubules and are at least 50% lower than the minimum classic frequency for infinitely long microtubules and could be one order of magnitude lower than the minimum classic frequency for shorter microtubules (only a few microns in length). These results suggest that the existing elastic foundation model may have overestimated the lowest frequencies of microtubules in vivo. Finally, based on our simulation results, some empirical relations are proposed for the critical (lowest) frequency of localized modes and the associated wave length. Compared to the classic elastic foundation model, the localized vibration modes and the associated wave lengths predicted by the present model are in better agreement with some known experimental observations.

The inhibition of aromatase alters the mechanical and rheological properties of non-small-cell lung cancer cell lines affecting cell migration

Tumor invasion and metastasis are key aspects of non-small cell lung cancer (NSCLC). During migration, cells undergo mechanical alterations. The mechanical phenotype of breast cancer cells is correlated with aromatase gene expression. We have previously shown that targeting aromatase is a promising strategy for NSCLC. The aim of this study was to examine morphological and mechanical changes of NSCLC cells, upon treatment with aromatase inhibitor and correlate their ability to migrate and invade. In vitro experiments were performed using H23 and A549 NSCLC cell lines and exemestane was used for aromatase inhibition. We demonstrated that exemestane reduced H23 cell migration and invasion and caused changes in cell morphology including increased vacuolar structures and greater pleomorphism. In addition, exemestane changed the distribution of α-tubulin in H23 and A549 cells in a way that might destabilize microtubules polymerization. These effects were associated with increased cell viscosity and decreased elastic shear modulus. Although exemestane caused similar effects in A549 cells regarding viscosity and elastic shear modulus, it did not affect A549 cell migration and caused an increase in invasion. The increased invasion was in line with vimentin perinuclear localization. Our data show that the treatment of NSCLC cells with an aromatase inhibitor not only affects cell migration and invasion but also alters the mechanical properties of the cells. It suggests that the different origin of cancer cells is associated with different morphological characteristics and mechanical behavior.

Tensegrity, cellular biophysics, and the mechanics of living systems

The recent convergence between physics and biology has led many physicists to enter the fields of cell and developmental biology. One of the most exciting areas of interest has been the emerging field of mechanobiology that centers on how cells control their mechanical properties, and how physical forces regulate cellular biochemical responses, a process that is known as mechanotransduction. In this article, we review the central role that tensegrity (tensional integrity) architecture, which depends on tensile prestress for its mechanical stability, plays in biology. We describe how tensional prestress is a critical governor of cell mechanics and function, and how use of tensegrity by cells contributes to mechanotransduction. Theoretical tensegrity models are also described that predict both quantitative and qualitative behaviors of living cells, and these theoretical descriptions are placed in context of other physical models of the cell. In addition, we describe how tensegrity is used at multiple size scales in the hierarchy of life—from individual molecules to whole living organisms—to both stabilize three-dimensional form and to channel forces from the macroscale to the nanoscale, thereby facilitating mechanochemical conversion at the molecular level.

A multiscale approach to modeling the passive mechanical contribution of cells in tissues

In addition to their obvious biological roles in tissue function, cells often play a significant mechanical role through a combination of passive and active behaviors. This study focused on the passive mechanical contribution of cells in tissues by improving our multiscale model via the addition of cells, which were treated as dilute spherical inclusions. The first set of simulations considered a rigid cell, with the surrounding ECM modeled as (1) linear elastic, (2) Neo-Hookean, and (3) a fiber network. Comparison with the classical composite theory for rigid inclusions showed close agreement at low cell volume fraction. The fiber network case exhibited nonlinear stress-strain behavior and Poisson's ratios larger than the elastic limit of 0.5, characteristics similar to those of biological tissues. The second set of simulations used a fiber network for both the cell (simulating cytoskeletal filaments) and matrix, and investigated the effect of varying relative stiffness between the cell and matrix, as well as the effect of a cytoplasmic pressure to enforce incompressibility of the cell. Results showed that the ECM network exerted negligible compression on the cell, even when the stiffness of fibers in the network was increased relative to the cell. Introduction of a cytoplasmic pressure significantly increased the stresses in the cell filament network, and altered how the cell changed its shape under tension. Findings from this study have implications on understanding how cells interact with their surrounding ECM, as well as in the context of mechanosensation.

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.

Actively stressed marginal networks

We study the effects of motor-generated stresses in disordered three-dimensional fiber networks using a combination of a mean-field theory, scaling analysis, and a computational model. We find that motor activity controls the elasticity in an anomalous fashion close to the point of marginal stability by coupling to critical network fluctuations. We also show that motor stresses can stabilize initially floppy networks, extending the range of critical behavior to a broad regime of network connectivities below the marginal point. Away from this regime, or at high stress, motors give rise to a linear increase in stiffness with stress. Finally, we demonstrate that our results are captured by a simple, constitutive scaling relation highlighting the important role of nonaffine strain fluctuations as a susceptibility to motor stress.

Substrate stiffness influences high resolution printing of living cells with an ink-jet system

The adaptation of inkjet printing technology for the realisation of controlled micro- and nano-scaled biological structures is of great potential in tissue and biomaterial engineering. In this paper we present the Olivetti BioJet system and its applications in tissue engineering and cell printing. BioJet, which employs a thermal inkjet cartridge, was used to print biomolecules and living cells. It is well known that high stresses and forces are developed during the inkjet printing process. When printing living particles (i.e., cell suspensions) the mechanical loading profile can dramatically damage the processed cells. Therefore computational models were developed to predict the velocity profile and the mechanical load acting on a droplet during the printing process. The model was used to investigate the role of the stiffness of the deposition substrate during droplet impact and compared with experimental investigations on cell viability after printing on different materials. The computational model and the experimental results confirm that impact forces are highly dependent on the deposition substrate and that soft and viscous surfaces can reduce the forces acting on the droplet, preventing cell damage. These results have high relevance for cell bioprinting; substrates should be designed to have a good compromise between substrate stiffness to conserve spatial patterning without droplet coalescence but soft enough to absorb the kinetic energy of droplets in order to maintain cell viability.

Response of an actin filament network model under cyclic stretching through a coarse grained Monte Carlo approach

Cells are complex, dynamic systems that actively adapt to various stimuli including mechanical alterations. Central to understanding cellular response to mechanical stimulation is the organization of the cytoskeleton and its actin filament network. In this manuscript, we present a minimalistic network Monte Carlo based approach to model actin filament organization under cyclic stretching. Utilizing a coarse-grained model, a filament network is prescribed within a two-dimensional circular space through nodal connections. When cyclically stretched, the model demonstrates that a perpendicular alignment of the filaments to the direction of stretch emerges in response to nodal repositioning to minimize net nodal forces from filament stress states. In addition, the filaments in the network rearrange and redistribute themselves to reduce the overall stress by decreasing their individual stresses. In parallel, we cyclically stretch NIH 3T3 fibroblasts and find a similar cytoskeletal response. With this work, we test the hypothesis that a first-principles mechanical model of filament assembly in a confined space is by itself capable of yielding the remodeling behavior observed experimentally. Identifying minimal mechanisms sufficient to reproduce mechanical influences on cellular structure has important implications in a diversity of fields, including biology, physics, medicine, computer science, and engineering.

Complexity of the tensegrity structure for dynamic energy and force distribution of cytoskeleton during cell spreading

Cytoskeleton plays important roles in intracellular force equilibrium and extracellular force transmission from/to attaching substrate through focal adhesions (FAs). Numerical simulations of intracellular force distribution to describe dynamic cell behaviors are still limited. The tensegrity structure comprises tension-supporting cables and compression-supporting struts that represent the actin filament and microtubule respectively, and has many features consistent with living cells. To simulate the dynamics of intracellular force distribution and total stored energy during cell spreading, the present study employed different complexities of the tensegrity structures by using octahedron tensegrity (OT) and cuboctahedron tensegrity (COT). The spreading was simulated by assigning specific connection nodes for radial displacement and attachment to substrate to form FAs. The traction force on each FA was estimated by summarizing the force carried in sounding cytoskeletal elements. The OT structure consisted of 24 cables and 6 struts and had limitations soon after the beginning of spreading by declining energy stored in struts indicating the abolishment of compression in microtubules. The COT structure, double the amount of cables and struts than the OT structure, provided sufficient spreading area and expressed similar features with documented cell behaviors. The traction force pointed inward on peripheral FAs in the spread out COT structure. The complex structure in COT provided further investigation of various FA number during different spreading stages. Before the middle phase of spreading (half of maximum spreading area), cell attachment with 8 FAs obtained minimized cytoskeletal energy. The maximum number of 12 FAs in the COT structure was required to achieve further spreading. The stored energy in actin filaments increased as cells spread out, while the energy stored in microtubules increased at initial spreading, peaked in middle phase, and then declined as cells reached maximum spreading. The dynamic flows of energy in struts imply that microtubules contribute to structure stabilization.

A multi-modular tensegrity model of an actin stress fiber

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

Propagation of mechanical stress through the actin cytoskeleton toward focal adhesions: model and experiment

We investigate both theoretically and experimentally how stress is propagated through the actin cytoskeleton of adherent cells and consequentially distributed at sites of focal adhesions (FAs). The actin cytoskeleton is modeled as a two-dimensional cable network with different lattice geometries. Both prestrain, resulting from actomyosin contractility, and central application of external force, lead to finite forces at the FAs that are largely independent of the lattice geometry, but strongly depend on the exact spatial distribution of the FAs. The simulation results compare favorably with experiments with adherent fibroblasts onto which lateral force is exerted using a microfabricated pillar. For elliptical cells, central application of external force along the long axis leads to two large stress regions located obliquely opposite to the pulling direction. For elliptical cells pulled along the short axis as well as for circular cells, there is only one region of large stress opposite to the direction of pull. If in the computer simulations FAs are allowed to rupture under force for elliptically elongated and circular cell shapes, then morphologies arise which are typical for migrating fibroblasts and keratocytes, respectively. The same effect can be obtained also by internally generated force, suggesting a mechanism by which cells can control their migration morphologies.

Tensegrity-based mechanosensing from macro to micro

This article is a summary of a lecture on cellular mechanotransduction that was presented at a symposium on "Cardiac Mechano-Electric Feedback and Arrhythmias" that convened at Oxford, England in April 2007. Although critical mechanosensitive molecules and cellular components, such as integrins, stretch-activated ion channels, and cytoskeletal filaments, have been shown to contribute to the response by which cells convert mechanical signals into a biochemical response, little is known about how they function in the structural context of living cells, tissues and organs to produce orchestrated changes in cell behavior in response to stress. Here, studies are reviewed that suggest our bodies use structural hierarchies (systems within systems) composed of interconnected extracellular matrix and cytoskeletal networks that span from the macroscale to the nanoscale to focus stresses on specific mechanotransducer molecules. A key feature of these networks is that they are in a state of isometric tension (i.e., experience a tensile prestress), which ensures that various molecular-scale mechanochemical transduction mechanisms proceed simultaneously and produce a concerted response. These features of living architecture are the same principles that govern tensegrity (tensional integrity) architecture, and mathematical models based on tensegrity are beginning to provide new and useful descriptions of living materials, including mammalian cells. This article reviews how the use of tensegrity at multiple size scales in our bodies guides mechanical force transfer from the macro to the micro, as well as how it facilitates conversion of mechanical signals into changes in ion flux, molecular binding kinetics, signal transduction, gene transcription, cell fate switching and developmental patterning.

Tools to study cell mechanics and mechanotransduction

Analysis of how cells sense and respond to mechanical stress has been limited by the availability of techniques that can apply controlled mechanical forces to living cells while simultaneously measuring changes in cell and molecular distortion, as well as alterations of intracellular biochemistry. We have confronted this challenge by developing new engineering methods to measure and manipulate the mechanical properties of cells and their internal cytoskeletal and nuclear frameworks, and by combining them with molecular cell biological techniques that rely on microscopic analysis and real-time optical readouts of biochemical signaling. In this chapter, we describe techniques like microcontact printing, magnetic twisting cytometry, and magnetic pulling cytometry that can be systematically used to study the molecular basis of cellular mechanotransduction.

Isostaticity and controlled force transmission in the cytoskeleton: A model awaiting experimental evidence

A new model is proposed for force transmission through the cytoskeleton (CSK). A general discussion is first presented on the physical principles that underlie the modeling of this phenomenon. Some fundamental problems of conventional models--continuous and discrete--are examined. It is argued that mediation of focused forces is essential for good control over intracellular mechanical signals. The difficulties of conventional continuous models in describing such mediation are traced to a fundamental assumption rather than to their being continuous. Relevant advantages and disadvantages of continuous and discrete modeling are discussed. It is concluded that favoring discrete models is based on two misconceptions, which are clarified. The model proposed here is based on the idea that focused propagation of mechanical stimuli in frameworks over large distances (compared to the mesh size) can only occur when considerable regions of the CSK are isostatic. The concept of isostaticity is explained and a recently developed continuous isostaticity theory is briefly reviewed. The model enjoys several advantages: it leads to good control over force mediation; it explains nonuniform stresses and action at a distance; it is continuous, making it possible to model force propagation over long distances; and it enables prediction of individual force paths. To be isostatic, or nearly so, CSK networks must possess specific structural characteristics, and these are quantified. Finally, several experimental observations are interpreted using the new model and implications are discussed. It is also suggested that this approach may give insight into the dynamics of reorganization of the CSK. Many of the results are amenable to experimental measurements, providing a testing ground for the proposed picture, and generic experiments are suggested.

Cellular mechanotransduction: putting all the pieces together again

Analysis of cellular mechanotransduction, the mechanism by which cells convert mechanical signals into biochemical responses, has focused on identification of critical mechanosensitive molecules and cellular components. Stretch-activated ion channels, caveolae, integrins, cadherins, growth factor receptors, myosin motors, cytoskeletal filaments, nuclei, extracellular matrix, and numerous other structures and signaling molecules have all been shown to contribute to the mechanotransduction response. However, little is known about how these different molecules function within the structural context of living cells, tissues, and organs to produce the orchestrated cellular behaviors required for mechanosensation, embryogenesis, and physiological control. Recent work from a wide range of fields reveals that organ, tissue, and cell anatomy are as important for mechanotransduction as individual mechanosensitive proteins and that our bodies use structural hierarchies (systems within systems) composed of interconnected networks that span from the macroscale to the nanoscale in order to focus stresses on specific mechanotransducer molecules. The presence of isometric tension (prestress) at all levels of these multiscale networks ensures that various molecular scale mechanochemical transduction mechanisms proceed simultaneously and produce a concerted response. Future research in this area will therefore require analysis, understanding, and modeling of tensionally integrated (tensegrity) systems of mechanochemical control.

Biomechanics of single zonal chondrocytes

Articular cartilage has a distinct zonal architecture, and previous work has shown that chondrocytes from different zones exhibit variations in gene expression and biosynthesis. In this study, the material properties of single chondrocytes from the superficial and middle/deep zones of bovine distal metatarsal articular cartilage were determined using unconfined compression and digital videocapture. To determine the viscoelastic properties of zonal chondrocytes, unconfined creep compression experiments were performed and the resulting creep curves of individual cells were fit using a standard linear viscoelastic solid model. In the model, a fixed value of the Poisson's ratio was used, determined optically from direct compression of middle/deep chondrocytes. The two approaches used in this study yielded the following average material properties of single chondrocytes: Poisson's ratio of 0.26+/-0.08, instantaneous modulus of 1.06+/-0.82 kPa, relaxed modulus of 0.78+/-0.58 kPa, and apparent viscosity of 4.08+/-7.20 kPa s. Superficial zone chondrocytes were found to be significantly stiffer than middle/deep zone chondrocytes. Attachment time did not affect the stiffness of the cells. The zonal variation in viscoelastic properties may result from the distinct mechanical environments experienced by the cells in vivo. Identifying intrinsic differences in the biomechanics of superficial and middle/deep zone chondrocytes is an important component in understanding how biomechanics influence articular cartilage health and disease.

Non-Hertzian Approach to Analyzing Mechanical Properties of Endothelial Cells Probed by Atomic Force Microscopy

Detailed measurements of cell material properties are required for understanding how cells respond to their mechanical environment. Atomic force microscopy (AFM) is an increasingly popular measurement technique that uniquely combines subcellular mechanical testing with high-resolution imaging. However, the standard method of analyzing AFM indentation data is based on a simplified “Hertz” theory that requires unrealistic assumptions about cell indentation experiments. The objective of this study was to utilize an alternative “pointwise modulus” approach, that relaxes several of these assumptions, to examine subcellular mechanics of cultured human aortic endothelial cells (HAECs). Data from indentations in 2‐to5‐μm square regions of cytoplasm reveal at least two mechanically distinct populations of cellular material. Indentations colocalized with prominent linear structures in AFM images exhibited depth-dependent variation of the apparent pointwise elastic modulus that was not observed at adjacent locations devoid of such structures. The average pointwise modulus at an arbitrary indentation depth of 200nm was 5.6±3.5kPa and 1.5±0.76kPa (mean±SD, n=7) for these two material populations, respectively. The linear structures in AFM images were identified by fluorescence microscopy as bundles of f-actin, or stress fibers. After treatment with 4μM cytochalasin B, HAECs behaved like a homogeneous linear elastic material with an apparent modulus of 0.89±0.46kPa. These findings reveal complex mechanical behavior specifically associated with actin stress fibers that is not accurately described using the standard Hertz analysis, and may impact how HAECs interact with their mechanical environment.

Effects of cytoskeletal prestress on cell rheological behavior

Normal tissue development requires that cells alter their mechanical behavior in different microenvironments to carry out their diverse functions. During cell spreading, migration, invasion and mitosis, cells exhibit a high degree of deformability, exhibiting almost a fluid-like behavior, whereas within quiescent differentiated tissues, cells must behave like an elastic solid to maintain their structural integrity in the face of an applied mechanical stress. A growing body of experimental evidence suggests that rheological properties of adherent cells depend on pre-existing tensional stress ("prestress") borne by the cytoskeleton. This prestress results from the action of tensional forces borne by actin microfilaments, transmitted over intermediate filaments and resisted by both extracellular matrix adhesions and internal microtubules. Observations that the prestress influences mechanical properties of the cell are intimately related to the cellular tensegrity model. This model depicts the cytoskeleton as an interconnected network of cables that carry pre-existing tension that is balanced by compression-bearing struts and by anchoring forces of the substrate. This paper offers a brief survey of the basic concept of cellular tensegrity model, comparison of model predictions with experimental data obtained from rheological measurements on living cells, and comparison with other models that have been used in studies of rheology of cells.

Analysis of Nonlinear Responses of Adherent Epithelial Cells Probed by Magnetic Bead Twisting: A Finite Element Model Based on a Homogenization Approach

An original homogenization method was used to analyze the nonlinear elastic properties of epithelial cells probed by magnetic twisting cytometry. In this approach, the apparent rigidity of a cell with nonlinear mechanical properties is deduced from the mechanical response of the entire population of adherent cells. The proposed hyperelastic cell model successfully accounts for the variability in probe-cell geometrical features, and the influence of the cell–substrate adhesion. Spatially distributed local secant elastic moduli had amplitudes ranging from 10 to 400 Pa. The nonlinear elastic behavior of cells may contribute to the wide differences in published results regarding cell elasticity moduli.

Mathematical model of retinal mosaic formation

The retina is one of the best examples of modular organisation in neural circuitry. This modular structure enables it to perform parallel processing. A mathematical model of the retina has been set up, focusing on mechanical features of retinal neurons and on the interaction and dendritic overlapping among retinal cells. The model focuses on the actions of local mechanical forces on the neuron's cytoskeleton. The cytoskeleton is regarded as a structure in which elastic and rigid elements are combined according to the tensegrity concept. We have assumed that dendritic overlap takes place in such a way as to favour uniform retinal neurons' distribution and that dendritic overlap is the only cause of neuron's motion on the retinal surface. This overlap depends on the growth of the dendrites due to the cytoskeletic deformation. The results obtained are in agreement with experimental results that support the notion that local mechanical interaction and dendritic overlapping are capable to transform random cell distributions into regular mosaics.

Mechanisms governing the visco-elastic responses of living cells assessed by foam and tensegrity models

The visco-elastic properties of living cells, measured to date by various authors, vary considerably, depending on the experimental methods and/or on the theoretical models used. In the present study, two mechanisms thought to be involved in cellular visco-elastic responses were analysed, based on the idea that the cytoskeleton plays a fundamental role in cellular mechanical responses. For this purpose, the predictions of an open unit-cell model and a 30-element visco-elastic tensegrity model were tested, taking into consideration similar properties of the constitutive F-actin. The quantitative predictions of the time constant and viscosity modulus obtained by both models were compared with previously published experimental data obtained from living cells. The small viscosity modulus values (10(0)-10(3) Pa x s) predicted by the tensegrity model may reflect the combined contributions of the spatially rearranged constitutive filaments and the internal tension to the overall cytoskeleton response to external loading. In contrast, the high viscosity modulus values (10(3)-10(5) Pa x s) predicted by the unit-cell model may rather reflect the mechanical response of the cytoskeleton to the bending of the constitutive filaments and/or to the deformation of internal components. The present results suggest the existence of a close link between the overall visco-elastic response of micromanipulated cells and the underlying architecture.

Mechanics of vimentin intermediate filaments

It is increasingly evident that the cytoskeleton of living cells plays important roles in mechanical and biological functions of the cells. Here we focus on the contribution of intermediate filaments (IFs) to the mechanical behaviors of living cells. Vimentin, a major structural component of IFs in many cell types, is shown to play an important role in vital mechanical and biological functions such as cell contractility, migration, stiffness, stiffening, and proliferation.

Tensegrity I. Cell structure and hierarchical systems biology

In 1993, a Commentary in this journal described how a simple mechanical model of cell structure based on tensegrity architecture can help to explain how cell shape, movement and cytoskeletal mechanics are controlled, as well as how cells sense and respond to mechanical forces (J. Cell Sci. 104, 613-627). The cellular tensegrity model can now be revisited and placed in context of new advances in our understanding of cell structure, biological networks and mechanoregulation that have been made over the past decade. Recent work provides strong evidence to support the use of tensegrity by cells, and mathematical formulations of the model predict many aspects of cell behavior. In addition, development of the tensegrity theory and its translation into mathematical terms are beginning to allow us to define the relationship between mechanics and biochemistry at the molecular level and to attack the larger problem of biological complexity. Part I of this two-part article covers the evidence for cellular tensegrity at the molecular level and describes how this building system may provide a structural basis for the hierarchical organization of living systems--from molecule to organism. Part II, which focuses on how these structural networks influence information processing networks, appears in the next issue.

Toward a generalised tensegrity model describing the mechanical behaviour of the cytoskeleton structure

The control of many cell functions including growth, migration and mechanotransduction, depends crucially on stress-induced mechanical changes in cell shape and cytoskeleton (CSK) structure. Quantitative studies have been carried out on 6-bar tensegrity models to analyse several mechanical parameters involved in the mechanical responses of adherent cells (i.e. strain hardening, internal stress and scale effects). In the present study, we attempt to generalize some characteristic mechanical laws governing spherical tensegrity structures, with a view of evaluating the mechanical behaviour of the hierarchical multi-modular CSK-structure. The numerical results obtained by studying four different tensegrity models are presented in terms of power laws and point to the existence of unique and constant relationships between the overall structural stiffness and the local properties (length, number and internal stress) of the constitutive components.

A prestressed cable network model of the adherent cell cytoskeleton

A prestressed cable network is used to model the deformability of the adherent cell actin cytoskeleton. The overall and microstructural model geometries and cable mechanical properties were assigned values based on observations from living cells and mechanical measurements on isolated actin filaments, respectively. The models were deformed to mimic cell poking (CP), magnetic twisting cytometry (MTC) and magnetic bead microrheometry (MBM) measurements on living adherent cells. The models qualitatively and quantitatively captured the fibroblast cell response to the deformation imposed by CP while exhibiting only some qualitative features of the cell response to MTC and MBM. The model for CP revealed that the tensed peripheral actin filaments provide the key resistance to indentation. The actin filament tension that provides mechanical integrity to the network was estimated at approximately 158 pN, and the nonlinear mechanical response during CP originates from filament kinematics. The MTC and MBM simulations revealed that the model is incomplete, however, these simulations show cable tension as a key determinant of the model response.

Cancer as a disease of epithelial-mesenchymal interactions and extracellular matrix regulation

Carcinogenesis - the process of cancer formation - is commonly discussed in terms of genetic alterations that lead to deregulation of cell growth. Recently, there has been a resurgence of interest in epigenetic factors and, in particular, the role of the stromal microenvironment and angiogenesis in tumor formation. In this article, cancer is presented as a disease of the developmental processes that govern how cells organize into tissues and tissues into organs. This histogenetic perspective raises the possibility that epithelial-mesenchymal interactions and the extracellular matrix (basement membrane) that is deposited through these interactions may actively contribute to the carcinogenic process. Experimental work is reviewed that confirms that extracellular matrix plays a key role in normal histodifferentiation during both epitheliogenesis and angiogenesis, and that epigenetic deregulation of cell-matrix interactions may actively promote tumor initiation and progression. The contributions of integrins, cytoskeleton, tensegrity and local variations in extracellular matrix mechanics to these processes are discussed, as are the implications of this work for future studies on cancer formation.

Mechanical signaling and the cellular response to extracellular matrix in angiogenesis and cardiovascular physiology

Great advances have been made in the identification of the soluble angiogenic factors, insoluble extracellular matrix (ECM) molecules, and receptor signaling pathways that mediate control of angiogenesis--the growth of blood capillaries. This review focuses on work that explores how endothelial cells integrate these chemical signals with mechanical cues from their local tissue microenvironment so as to produce functional capillary networks that exhibit specialized form as well as function. These studies have revealed that ECM governs whether an endothelial cell will switch between growth, differentiation, motility, or apoptosis programs in response to a soluble stimulus based on its ability to mechanically resist cell tractional forces and thereby produce cell and cytoskeletal distortion. Transmembrane integrin receptors play a key role in this mechanochemical transduction process because they both organize a cytoskeletal signaling complex within the focal adhesion and preferentially focus mechanical forces on this site. Molecular filaments within the internal cytoskeleton--microfilaments, microtubules, and intermediate filaments--also contribute to the cell's structural and functional response to mechanical stress through their role as discrete support elements within a tensegrity-stabilized cytoskeletal array. Importantly, a similar form of mechanical control also has been shown to be involved in the regulation of contractility in vascular smooth muscle cells and cardiac myocytes. Thus, the mechanism by which cells perform mechanochemical transduction and the implications of these findings for morphogenetic control are discussed in the wider context of vascular development and cardiovascular physiology.

A cellular tensegrity model to analyse the structural viscoelasticity of the cytoskeleton

This study describes the viscoelastic properties of a refined cellular-tensegrity model composed of six rigid bars connected to a continuous network of 24 viscoelastic pre-stretched cables (Voigt bodies) in order to analyse the role of the cytoskeleton spatial rearrangement on the viscoelastic response of living adherent cells. This structural contribution was determined from the relationships between the global viscoelastic properties of the tensegrity model, i.e., normalized viscosity modulus (eta(*)), normalized elasticity modulus (E(*)), and the physical properties of the constitutive elements, i.e., their normalized length (L(*)) and normalized initial internal tension (T(*)). We used a numerical method to simulate the deformation of the structure in response to different types of loading, while varying by several orders of magnitude L(*) and T(*). The numerical results obtained reveal that eta(*) remains almost independent of changes in T(*) (eta(*) proportional, variant T(*+0.1)), whereas E(*) increases with approximately the square root of the internal tension T(*) (from E(*) proportional, variant T(*+0.3) to E(*) proportional, variant T(*+0.7)). Moreover, structural viscosity eta(*) and elasticity E(*) are both inversely proportional to the square of the size of the structure (eta(*) proportional, variant L(*-2) and E(*) proportional, variant L(*-2)). These structural properties appear consistent with cytoskeleton (CSK) mechanical properties measured experimentally by various methods which are specific to the CSK micromanipulation in living adherent cells. Present results suggest, for the first time, that the effect of structural rearrangement of CSK elements on global CSK behavior is characterized by a faster cellular mechanical response relatively to the CSK element response, which thus contributes to the solidification process observed in adherent cells. In extending to the viscoelastic properties the analysis of the mechanical response of the cellular 30-element tensegrity model, the present study contributes to the understanding of recent results on the cellular-dynamic response and allows to reunify the scattered data reported for the viscoelastic properties of living adherent cells.

Assessment of mechanical properties of adherent living cells by bead micromanipulation: comparison of magnetic twisting cytometry vs optical tweezers

We compare the measurements of viscoelastic properties of adherent alveolar epithelial cells by two micromanipulation techniques: (i) magnetic twisting cytometry and (ii) optical tweezers, using microbeads of same size and similarly attached to F-actin. The values of equivalent Young modulus E, derived from linear viscoelasticity theory, become consistent when the degree of bead immersion in the cell is taken into account. E-values are smaller in (i) than in (ii): approximately 34-58 Pa vs approximately 29-258 Pa, probably because higher stress in (i) reinforces nonlinearity and cellular plasticity. Otherwise, similar relaxation time constants, around 2 s, suggest similar dissipative mechanisms.

Three-dimensional cellular deformation analysis with a two-photon magnetic manipulator workstation

The ability to apply quantifiable mechanical stresses at the microscopic scale is critical for studying cellular responses to mechanical forces. This necessitates the use of force transducers that can apply precisely controlled forces to cells while monitoring the responses noninvasively. This paper describes the development of a micromanipulation workstation integrating two-photon, three-dimensional imaging with a high-force, uniform-gradient magnetic manipulator. The uniform-gradient magnetic field applies nearly uniform forces to a large cell population, permitting statistical quantification of select molecular responses to mechanical stresses. The magnetic transducer design is capable of exerting over 200 pN of force on 4.5-microm-diameter paramagnetic particles and over 800 pN on 5.0-microm ferromagnetic particles. These forces vary within +/-10% over an area 500 x 500 microm2. The compatibility with the use of high numerical aperture (approximately 1.0) objectives is an integral part of the workstation design allowing submicron-resolution, three-dimensional, two-photon imaging. Three-dimensional analyses of cellular deformation under localized mechanical strain are reported. These measurements indicate that the response of cells to large focal stresses may contain three-dimensional global deformations and show the suitability of this workstation to further studying cellular response to mechanical stresses.

Cell prestress. I. Stiffness and prestress are closely associated in adherent contractile cells

The tensegrity hypothesis holds that the cytoskeleton is a structure whose shape is stabilized predominantly by the tensile stresses borne by filamentous structures. Accordingly, cell stiffness must increase in proportion with the level of the tensile stress, which is called the prestress. Here we have tested that prediction in adherent human airway smooth muscle (HASM) cells. Traction microscopy was used to measure the distribution of contractile stresses arising at the interface between each cell and its substrate; this distribution is called the traction field. Because the traction field must be balanced by tensile stresses within the cell body, the prestress could be computed. Cell stiffness (G) was measured by oscillatory magnetic twisting cytometry. As the contractile state of the cell was modulated with graded concentrations of relaxing or contracting agonists (isoproterenol or histamine, respectively), the mean prestress ((t)) ranged from 350 to 1,900 Pa. Over that range, cell stiffness increased linearly with the prestress: G (Pa) = 0.18(t) + 92. While this association does not necessarily preclude other interpretations, it is the hallmark of systems that secure shape stability mainly through the prestress. Regardless of mechanism, these data establish a strong association between stiffness of HASM cells and the level of tensile stress within the cytoskeleton.