Interrelations between elastic energy and strain in a tensegrity model: contribution to the analysis of the mechanical response in living cells

Interactions between the physical and physiological properties of cellular sub-units result in changes in the shape and mechanical behaviour of living tissues. To understand the mechanotransmission processes, models are needed to describe the complex interrelations between the elements and the cytoskeletal structure. In this study, we used a 30-element tensegrity structure to analyse the influence of the type of loading on the mechanical response and shape changes of the cell. Our numerical results, expressed in terms of strain energy as a function of the overall deformation of the tensegrity structure, suggest that changes in cell functions during mechanical stimuli for a given potential energy are correlated to the type of loading applied, which determines the resultant changes in cell shape. The analysis of these cellular deformations may explain the large variability in the response of bone cells submitted to different types of mechanical loading.     

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.     

Tensegrity architecture explains linear stiffening and predicts softening of living cells

The problem of theoretical explanation of the experimentally observed linear stiffening of living cells is addressed. This explanation is based on Ingber's assumption that the cell cytoskeleton, which enjoys tensegrity architecture with compressed microtubules that provide tension to the microfilaments, affects the mechanical behavior of the living cell. Moreover, it is shown that the consideration of the extreme flexibility of microtubules and the unilateral response of microfilaments is crucial for the understanding of the living cell overall behavior. Formal nonlinear structural analysis of the cell cytoskeleton under external mechanical loads is performed. For this purpose, a general computer model for tensegrity assemblies with unilateral microfilaments and buckled microtubules is developed and applied to the theoretical analysis of the mechanical response of 2D and 3D examples of tensegrity cells mimicking the behavior of real living cells. Results of the computer simulations explain the experimentally observed cell stiffening. Moreover, the theoretical results predict the possible existence of a transient softening behavior of cells, a phenomenon, which has not been observed in experiments yet.     

On tensegrity in cell mechanics

All models are wrong, but some are useful. This famous saying mirrors the situation in cell mechanics as well. It looks like no particular model of the cell deformability can be unconditionally preferred over others and different models reveal different aspects of the mechanical behavior of living cells. The purpose of the present work is to discuss the so-called tensegrity models of the cell cytoskeleton. It seems that the role of the cytoskeleton in the overall mechanical response of the cell was not appreciated until Donald Ingber put a strong emphasis on it. It was fortunate that Ingber linked the cytoskeletal structure to the fascinating art of tensegrity architecture. This link sparked interest and argument among biologists, physicists, mathematicians, and engineers. At some point the enthusiasm regarding tensegrity perhaps became overwhelming and as a reaction to that some skepticism built up. To demystify Ingber's ideas the present work aims at pinpointing the meaning of tensegrity and its role in our understanding of the importance of the cytoskeleton for the cell deformability and motility. It should be noted also that this paper emphasizes basic ideas rather than carefully follows the chronology of the development of tensegrity models. The latter can be found in the comprehensive review by Dimitrije Stamenovic (2006) to which the present work is complementary.     

Cell cytoskeleton and tensegrity

The role of tensegrity architecture of the cytoskeleton in the mechanical behavior of living cells is examined by computational studies. Plane and spatial tensegrity models of the cytoskeleton are considered as well as their non-tensegrity counterparts. Local buckling including deep postbuckling response of the compressed microtubules of the cytoskeleton is considered. The tensioned microfilaments cannot sustain compression. Large deformation of the whole model is accounted and fully nonlinear analysis is performed. It is shown that in the case of local buckling of the microtubules non-tensegrity models exhibit qualitatively the same linear stiffening as their tensegrity counterparts. This result raises the question of experimental validation of the local buckling of microtubules. If the microtubules of real cells are not straight, then tensegrity (in a narrow sense) is not a necessary attribute of the cytoskeleton architecture. If the microtubules are straight then tensegrity is more likely to be the cytoskeletal architecture.     

Hydrostatic pressure sensation in cells: integration into the tensegrity model.

Hydrostatic pressure (HP) is a mechanical stimulus that has received relatively little attention in the field of the cell biology of mechanotransduction. Generalized models, such as the tensegrity model, do not provide a detailed explanation of how HP might be detected. This is significant, because HP is an important mechanical stimulus, directing cell behaviour in a variety of tissues, including cartilage, bone, airways, and the vasculature. HP sensitivity may also be an important factor in certain clinical situations, as well as under unique environmental conditions such as microgravity. While downstream cellular effects have been well characterized, the initial HP sensation mechanism remains unclear. In vitro evidence shows that HP affects cytoskeletal polymerization, an effect that may be crucial in triggering the cellular response. The balance between free monomers and cytoskeletal polymers is shifted by alterations in HP, which could initiate a cellular response by releasing and (or) activating cytoskeleton-associated proteins. This new model fits well with the basic tenets of the existing tensegrity model, including mechanisms in which cellular HP sensitivity could be tuned to accommodate variable levels of stress.     

Spatial mutation of the T cell immunological synapse

The hardware for intracellular signaling networks consists of cascades of chemical reactions. It is becoming increasingly apparent that the large-scale spatial organization of molecules in these networks can lead to differential outcomes from otherwise chemically equivalent systems. This has amplified interest in controlled spatial organization as a regulator of cellular signal transduction. In response, a new category of experimentation is developing, in which the spatial positions of signaling molecules in living cells are directly manipulated through mechanical means. These methodologies complement conventional genetic and pharmacological approaches, both of which are chemical in nature, by perturbing the system through exclusively physical mechanisms.     

Tensegrity behaviour of cortical and cytosolic cytoskeletal components in twisted living adherent cells

The present study is an attempt to relate the multicomponent response of the cytoskeleton (CSK), evaluated in twisted living adherent cells, to the heterogeneity of the cytoskeletal structure - evaluated both experimentally by means of 3D reconstructions, and theoretically considering the predictions given by two tensegrity models composed of (four and six) compressive elements and (respectively 12 and 24) tensile elements. Using magnetic twisting cytometry in which beads are attached to integrin receptors linked to the actin CSK of living adherent epithelial cells, we specifically measured the elastic CSK response at quasi equilibrium state and partitioned this response in terms of cortical and cytosolic contributions with a two-component model (i.e., a series of two Voigt bodies). These two CSK components were found to be prestressed and exhibited a stress-hardening response which both characterize tensegrity behaviour with however significant differences: compared to the cytosolic component, the cortical cytoskeleton appears to be a faster responding component, being a less prestressed and easily deformable structure. The discrepancies in elastic behaviour between the cortical and cytosolic CSK components may be understood on the basis of prestress tensegrity model predictions, given that the length and number of constitutive actin elements are taken into account.     

The origin of cellular life

This essay presents a scenario of the origin of life that is based on analysis of biological architecture and mechanical design at the microstructural level. My thesis is that the same architectural and energetic constraints that shape cells today also guided the evolution of the first cells and that the molecular scaffolds that support solid-phase biochemistry in modern cells represent living microfossils of past life forms. This concept emerged from the discovery that cells mechanically stabilize themselves using tensegrity architecture and that these same building rules guide hierarchical self-assembly at all size scales (Sci. Amer 278:48-57;1998). When combined with other fundamental design principles (e.g., energy minimization, topological constraints, structural hierarchies, autocatalytic sets, solid-state biochemistry), tensegrity provides a physical basis to explain how atomic and molecular elements progressively self-assembled to create hierarchical structures with increasingly complex functions, including living cells that can self-reproduce.     

Combining mechanical and optical approaches to dissect cellular mechanobiology

Mechanical force modulates a wide array of cell physiological processes. Cells sense and respond to mechanical stimuli using a hierarchy of structural complexes spanning multiple length scales, including force-sensitive molecules and cytoskeletal networks. Understanding mechanotransduction, i.e., the process by which cells convert mechanical inputs into biochemical signals, has required the development of novel biophysical tools that allow for probing of cellular and subcellular components at requisite time, length, and force scales and technologies that track the spatio-temporal dynamics of relevant biomolecules. In this review, we begin by discussing the underlying principles and recent applications of atomic force microscopy, magnetic twisting cytometry, and traction force microscopy, three tools that have been widely used for measuring the mechanical properties of cells and for probing the molecular basis of cellular mechanotransduction. We then discuss how such tools can be combined with advanced fluorescence methods for imaging biochemical processes in living cells in the context of three specific problem spaces. We first focus on fluorescence resonance energy transfer, which has enabled imaging of intra- and inter-molecular interactions and enzymatic activity in real time based on conformational changes in sensor molecules. Next, we examine the use of fluorescence methods to probe force-dependent dynamics of focal adhesion proteins. Finally, we discuss the use of calcium ratiometric signaling to track fast mechanotransductive signaling dynamics. Together, these studies demonstrate how single-cell biomechanical tools can be effectively combined with molecular imaging technologies for elucidating mechanotransduction processes and identifying mechanosensitive proteins.     

Single cell mechanotransduction and its modulation analyzed by atomic force microscope indentation

The skeleton adapts to its mechanical usage, although at the cellular level, the distribution and magnitude of strains generated and their detection are ill-understood. The magnitude and nature of the strains to which cells respond were investigated using an atomic force microscope (AFM) as a microindentor. A confocal microscope linked to the setup enabled analysis of cellular responses. Two different cell response pathways were identified: one, consequent upon contact, depended on activation of stretch-activated ion channels; the second, following stress relaxation, required an intact microtubular cytoskeleton. The cellular responses could be modulated by selectively disrupting cytoskeletal components thought to be involved in the transduction of mechanical stimuli. The F-actin cytoskeleton was not required for responses to mechanical strain, whereas the microtubular and vimentin networks were. Treatments that reduced membrane tension, or its transmission, selectively reduced contact reactions. Immunostaining of the cell cytoskeleton was used to interpret the results of the cytoskeletal disruption studies. We provide an estimate of the cellular strain magnitude needed to elicit intracellular calcium responses and propose a model that links single cell responses to whole bone adaptation. This technique may help to understand adaptation to mechanical usage in other organs.     

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.     

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.     

Models of cytoskeletal mechanics of adherent cells

 Adherent cells sense their mechanical environment, which, in turn, regulates their functions. During the past decade, a growing body of evidence has indicated that a deformable, solid-state intracellular structure known as the cytoskeleton (CSK) plays a major role in transmitting and distributing mechanical stresses within the cell as well as in their conversion into a chemical response. Therefore in order to understand mechanical regulation and control of cellular functions, one needs to understand mechanisms that determine how the CSK changes its shape and mechanics in response to stress. In this survey, we examined commonly used structurally based models of the CSK. In particular, we focused on two classes of these models: open-cell foam networks and stress-supported structures. We identified the underlying mechanisms that determine deformability of those models and compare model predictions with data previously obtained from mechanical tests on cultured living adherent cells at steady state. We concluded that stress-supported structures appear more suitable for describing cell deformability because this class of structures can explain the central role that the cytoskeletal contractile prestress plays in cellular mechanics. Received: 2 January 2002 / Accepted: 27 February 2002     

Brushes,cables, and anchors: Recent insights into multiscale assembly and mechanics of cellular structural networks

The remarkable ability of living cells to sense, process, and respond to mechanical stimuli in their environment depends on the rapid and efficient interconversion of mechanical and chemical energy at specific times and places within the cell. For example, application of force to cells leads to conformational changes in specific mechanosensitive molecules which then trigger cellular signaling cascades that may alter cellular structure, mechanics, and migration and profoundly influence gene expression. Similarly, the sensitivity of cells to mechanical stresses is governed by the composition, architecture, and mechanics of the cellular cytoskeleton and extracellular matrix (ECM), which are in turn driven by molecular-scale forces between the constituent biopolymers. Understanding how these mechanochemical systems coordinate over multiple length and time scales to produce orchestrated cell behaviors represents a fundamental challenge in cell biology. Here, we review recent advances in our understanding of these complex processes in three experimental systems: the assembly of axonal neurofilaments, generation of tensile forces by actomyosin stress fiber bundles, and mechanical control of adhesion assembly.     

Signal transduction by integrin receptors for extracellular matrix: cooperative processing of extracellular information.

Adhesion receptors allow cells to interact with a dynamic and information-rich environment of extracellular matrix molecules. The integrin family of adhesion receptors transduces signals from the extracellular matrix that regulate growth, gene expression and differentiation, as well as cell shape, motility and cytoskeletal architecture. Recent data support the hypothesis that integrins transduce signals cooperatively with other classes of adhesion receptors or with growth factor receptors. Furthermore, the ability of integrins to interact with the cytoskeleton appears to be fundamental to their mechanism for signal transduction.     

Imaging and manipulating the structural machinery of living cells on the micro- and nanoscale

The structure, physiology, and fate of living cells are all highly sensitive to mechanical forces in the cellular microenvironment, including stresses and strains that originate from encounters with the extracellular matrix (ECM), blood and other flowing materials, and neighbouring cells. This relationship between context and physiology bears tremendous implications for the design of cellular micro-or nanotechnologies, since any attempt to control cell behavior in a device must provide the appropriate physical microenvironment for the desired cell behavior. Cells sense, process, and respond to biophysical cues in their environment through a set of integrated, multi-scale structural complexes that span length scales from single molecules to tens of microns, including small clusters of force-sensing molecules at the cell surface, micron-sized cell-ECM focal adhesion complexes, and the cytoskeleton that permeates and defines the entire cell. This review focuses on several key technologies that have recently been developed or adapted for the study of the dynamics of structural micro-and nanosystems in living cells and how these systems contribute to spatially-and temporally-controlled changes in cellular structure and mechanics. We begin by discussing subcellular laser ablation, which permits the precise incision of nanoscale structural elements in living cells in order to discern their mechanical properties and contributions to cell structure. We then discuss fluorescence recovery after photobleaching and fluorescent speckle microscopy, two live-cell fluorescence imaging methods that enable quantitative measurement of the binding and transport properties of specific proteins in the cell. Finally, we discuss methods to manipulate cellular structural networks by engineering the extracellular environment, including microfabrication of ECM distributions of defined geometry and microdevices designed to measure cellular traction forces at micron-scale resolution. Together, these methods form a powerful arsenal that is already adding significantly to our understanding of the nanoscale architecture and mechanics of living cells and may contribute to the rational design of new cellular micro-and nanotechnologies.     

Directional dependence of osteoblastic calcium response to mechanical stimuli

In adaptive bone remodeling, mechanical signals such as stress/strain caused by loading/deformation are believed to play important roles as regulators of the process in which osteoclastic resorption and osteoblastic formation are coordinated under a local mechanical environment. The mechanism by which cells sense and transduce mechanical signals to the intracellular biochemical signaling cascade is still unclear, however to address this issue, the present study investigated the characteristic response of a single osteoblastic cell, MC3T3-E1, to a well-defined mechanical stimulus and the involvement of the cytoskeletal actin fiber structure in the mechanotransduction pathway. First, by mechanically perturbing to a single cell using a microneedle, a change in the intracellular calcium ion concentration [Ca²⁺]_i was observed as a primal signaling response to a mechanical stimulus, and the threshold value of the perturbation as the mechanical stimulus was evaluated quantitatively. Second, to study directional dependence of the response to the mechanical stimulus, the effect of actin fiber orientation on the threshold value of the calcium response was investigated at various magnitudes and directions of the stimulus. It was found that the osteoblastic response to the perturbation exhibited a directional dependence. That is, the sensitivity of osteoblastic cells to a mechanical stimulus depends on the angle of the applied deformation with respect to the cytoskeletal actin fiber orientation. This finding is phenomenological evidence that cytoskeletal actin fiber structures are involved in the mechanotransduction mechanism, which may be related to cell polarization behaviors such as cellular alignment caused by mechanical stimulation.     

Shape-dependent control of cell growth, differentiation, and apoptosis: switching between attractors in cell regulatory networks

Development of characteristic tissue patterns requires that individual cells be switched locally between different phenotypes or "fates;" while one cell may proliferate, its neighbors may differentiate or die. Recent studies have revealed that local switching between these different gene programs is controlled through interplay between soluble growth factors, insoluble extracellular matrix molecules, and mechanical forces which produce cell shape distortion. Although the precise molecular basis remains unknown, shape-dependent control of cell growth and function appears to be mediated by tension-dependent changes in the actin cytoskeleton. However, the question remains: how can a generalized physical stimulus, such as cell distortion, activate the same set of genes and signaling proteins that are triggered by molecules which bind to specific cell surface receptors. In this article, we use computer simulations based on dynamic Boolean networks to show that the different cell fates that a particular cell can exhibit may represent a preprogrammed set of common end programs or "attractors" which self-organize within the cell's regulatory networks. In this type of dynamic network model of information processing, generalized stimuli (e.g., mechanical forces) and specific molecular cues elicit signals which follow different trajectories, but eventually converge onto one of a small set of common end programs (growth, quiescence, differentiation, apoptosis, etc.). In other words, if cells use this type of information processing system, then control of cell function would involve selection of preexisting (latent) behavioral modes of the cell, rather than instruction by specific binding molecules. Importantly, the results of the computer simulation closely mimic experimental data obtained with living endothelial cells. The major implication of this finding is that current methods used for analysis of cell function that rely on characterization of linear signaling pathways or clusters of genes with common activity profiles may overlook the most critical features of cellular information processing which normally determine how signal specificity is established and maintained in living cells.