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.     

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.     

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.     

Probing cell structure by controlling the mechanical environment with cell–substrate interactions

Recent results demonstrate the exquisite sensitivity of cell morphology and structure to mechanical stimulation. Mechanical stimulation is often coupled with cell–substrate interactions that can, in turn, influence molecular response and determine cellular fates including apoptosis, proliferation, and differentiation. To understand these effects as they specifically relate to compressive mechanical stimulation and topographic control, we developed a microfabricated system to grow cells on polydimethylsiloxane (PDMS) microchannel surfaces where we maintained compression stimulation. We also probed cellular response following compressive mechanical stimulation to PDMS substrates of varying stiffness. In these instances, we examined cytoskeletal and morphologic changes in living cells attached to our substrate following the application of localized compressive stimulation. We found that the overall morphology and cell structure, including the actin cytoskeleton, oriented in the direction of the compressive strain applied and along the topographic microchannels. Furthermore by comparing topographic response to material stiffness, we found a 40% increase in cell area for cells cultured on the microchannels versus softer PDMS as well as a decreased cell area of 30% when using softer PDMS over unmodified PDMS. These findings have implications for research in a diversity of fields including cell–material interactions, mechanotransduction, and tissue engineering.     

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.     

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.     

Theoretical study of intracellular stress fiber orientation under cyclic deformation

We studied stress fiber orientation under a wide range of uniaxial cyclic deformations. We devised and validated a hypothesis consisting of two parts, as follows: (1) a stress fiber aligns to avoid a mechanical stimulus in the fiber direction under cyclic deformation. This means that, among all allowable directions, a stress fiber aligns in the direction which minimizes the stimulus, i. e., the summation of the changes in length of the stress fiber over one stretch cycle; and (2) there is a limit in the sensitivity of the cellular response to the mechanical stimulus. Due to this sensing limit, the orientation angle in stress fibers is distributed around the angle corresponding to the minimum stimulus. To validate this hypothesis, we approximated an anisotropic deformation of the membrane on which cells were to be cultured. We then obtained the relationships between the stretch range and the fiber angle in the undeformed state which minimize the mechanical stimuli, assuming that the membrane on which stress fibers and cells adhered was homogeneous and incompressible. Numerical simulation results showed that the proposed hypothesis described our previous experimental results well and was consistent with the experimental results in the literature. The simulation results, taking account of the second part of the hypothesis with a small value for the limit in sensitivity to the mechanical stimulus, could explain why cell orientation is distributed so widely with cyclic stretch ranges of <10%. The proposed hypothesis can be applied to various types of deformation because the mechanical stimulus is always sensed and accumulates under cyclic deformation without the necessity of a reference state to measure the stimulus.     

Contractile equilibration of single cells to step changes in extracellular stiffness

Extracellular stiffness has been shown to alter long timescale cell behaviors such as growth and differentiation, but the cellular response to changes in stiffness on short timescales is poorly understood. By studying the contractile response of cells to dynamic stiffness conditions using an atomic force microscope, we observe a seconds-timescale response to a step change in extracellular stiffness. Specifically, we observe acceleration in contraction velocity (μm/min) and force rate (nN/min) upon a step decrease in stiffness and deceleration upon a step increase in stiffness. Interestingly, this seconds-timescale response to a change in extracellular stiffness is not altered by inhibiting focal adhesion signaling or stretch-activated ion channels and is independent of cell height and contraction force. Rather, the response timescale is altered only by disrupting cytoskeletal mechanics and is well described by a simple mechanical model of a constant velocity actuator pulling against an internal cellular viscoelastic network. Consistent with the predictions of this model, we find that an osmotically expanding hydrogel responds to step changes in extracellular stiffness in a similar manner to cells. We therefore propose that an initial event in stiffness sensing is establishment of a mechanical equilibrium that balances contraction of the viscoelastic cytoskeleton with deformation of the extracellular matrix.     

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.     

Stiffening response of a cellular tensegrity model

Living cells exhibit, as most biological tissues, a stiffening (strain-hardening) response which reflects the nonlinearity of the stress-strain relationship. Tensegrity structures have been proposed as a comprehensive model of such a cell's mechanical response. Based on a theoretical model of a 30-element tensegrity structure, we propose a quantitative analysis of its nonlinear mechanical behavior under static conditions and large deformations. This study provides theoretical foundation to the passage from large-scale tensegrity models to microscale living cells, as well as the comparison between results obtained in biological specimens of different sizes. We found two non-dimensional parameters (L*-normalized element length and T*-normalized elastic tension) which govern the mechanical response of the structure for three types of loading tested (extension, compression and shear). The linear strain-hardening is uniquely observed for extension but differed for the two other types of loading tested. The stiffening response of the theoretical model was compared and discussed with the living cells stiffening response observed by different methods (shear flow experiments, micromanipulation and magnetocytometry).     

The Poisson Ratio of the Cellular Actin Cortex Is Frequency Dependent

Cell shape changes are vital for many physiological processes such as cell proliferation, cell migration, and morphogenesis. They emerge from an orchestrated interplay of active cellular force generation and passive cellular force response, both crucially influenced by the actin cytoskeleton. To model cellular force response and deformation, cell mechanical models commonly describe the actin cytoskeleton as a contractile isotropic incompressible material. However, in particular at slow frequencies, there is no compelling reason to assume incompressibility because the water content of the cytoskeleton may change. Here, we challenge the assumption of incompressibility by comparing computer simulations of an isotropic actin cortex with tunable Poisson ratio to measured cellular force response. Comparing simulation results and experimental data, we determine the Poisson ratio of the cortex in a frequency-dependent manner. We find that the Poisson ratio of the cortex decreases in the measured frequency regime analogous to trends reported for the Poisson ratio of glassy materials. Our results therefore indicate that actin cortex compression or dilation is possible in response to acting forces at sufficiently fast timescales. This finding has important implications for the parameterization in active gel theories that describe actin cytoskeletal dynamics.     

Mechanical control of tissue morphogenesis during embryological development

Twenty years ago, we proposed a model of developmental control based on tensegrity architecture, in which tissue pattern formation in the embryo is controlled through mechanical interactions between cells and extracellular matrix (ECM) which place the tissue in a state of isometric tension (prestress). The model proposed that local changes in the mechanical compliance of the ECM, for example, due to regional variations in basement membrane degradation beneath growing epithelium, may result in local stretching of the ECM and associated adherent cells, much like a "run-in-a-stocking". Cell growth and function would be controlled locally though physical distortion of the associated cells, or changes in cytoskeletal tension. Importantly, experimental studies have demonstrated that cultured cells can be switched between different fates, including growth, differentiation, apoptosis, directional motility and different stem cell lineages, by modulating cell shape. Experiments in whole embryonic organ rudiments also have confirmed the tight correlation between basement membrane thinning, cell tension generation and new bud and branch formation during tissue morphogenesis and that this process can be inhibited or accelerated by dissipating or enhancing cytoskeletal tension, respectively. Taken together, this work confirms that mechanical forces generated in the cytoskeleton of individual cells and exerted on ECM scaffolds, play a critical role in the sculpting of the embryo.     

Microtubules may harden or soften cells, depending of the extent of cell distension

Experimental data show that disruption of microtubules causes cells to either become stiffer or softer. Current understanding of these behaviors is based on several different mechanisms, each of which can account for only stiffening or softening. In this study we offer a model that can explain both these features. The model is based on the cellular tensegrity idea. Key premises of the model are that cell shape stability is secured through pre-existing mechanical stress (prestress) borne by the actin cytoskeletal network, and that this prestress is partly balanced by cytoskeletal microtubules and partly by the extracellular matrix. Thus, disturbance of this balance would affect cell deformability. The model predicts that disruption of microtubules causes an increase or a decrease in cell stiffness, depending on the extent to which microtubules participate in balancing the prestress which, in turn, depends on the extent of cell spreading. In highly spread cells microtubules have a minor and negative contribution to cell stiffness, whereas in less spread cells their contribution is positive and substantial. Since in their natural habitat cells seldom exhibit highly spread forms, the above results suggest that the contribution of microtubules to cell deformability cannot be overlooked.     

Measurement of local strain on cell membrane at initiation point of calcium signaling response to applied mechanical stimulus in osteoblastic cells

In adaptive bone remodeling, it is believed that bone cells such as osteoblasts, osteocytes and osteoclasts can sense mechanical stimuli and modulate their remodeling activities. However, the mechanosensing mechanism by which these cells sense mechanical stimuli and transduce mechanical signals into intracellular biochemical signals is still not clearly understood. From the viewpoint of cell biomechanics, it is important to clarify the mechanical conditions under which the cellular mechanosensing mechanism is activated. The aims of this study were to evaluate a mechanical condition, that is, the local strain on the cell membrane, at the initiation point of the intracellular calcium signaling response to the applied mechanical stimulus in osteoblast-like MC3T3-E1 cells, and to investigate the effect of deformation velocity on the characteristics of the cellular response. To apply a local deformation to a single cell, a glass microneedle was directly indented to the cell and moved horizontally on the cell membrane. To observe the cellular response and the deformation of the cell membrane, intracellular calcium ions and the cell membrane were labeled using fluorescent dyes and simultaneously observed by confocal laser scanning microscopy. The strain distribution on the cell membrane attributable to the applied local deformation and the strain magnitude at the initiation point of the calcium signaling responses were analyzed using obtained fluorescence images. From two-dimensionally projected images, it was found that there is a local compressive strain at the initiation point of calcium signaling. Moreover, the cellular response revealed velocity dependence, that is, the cells seemed to respond with a higher sensitivity to a higher deformation velocity. From the viewpoint of cell biomechanics, these results provide us a fundamental understanding of the mechanosensing mechanism of osteoblast-like cells.     

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.     

Experimental tests of the cellular tensegrity hypothesis

The tensegrity model depicts the cytoskeleton (CSK) as a prestressed network of interconnected filaments. The prestress is generated by the CSK contractile apparatus and is partly balanced by traction at the cell-substrate interface and partly by CSK internal compression elements such as microtubules (MTs). A key feature of tensegrity is that the shear modulus (G) must increase in proportion with the prestress. Here we have tested that prediction as well as the idea that compression of MTs balance a portion of the cell prestress. Airway smooth muscle cells were studied. Traction microscopy was used to calculate traction. Because traction must be balanced by the stress within the cell, the prestress could be computed. Cell G was measured by oscillatory magnetic cytometry. The prestress was modulated using graded concentrations of contracting (histamine) or relaxing (isoproterenol) agonists and by disrupting MTs by colchicine. It was found that G increased in proportion with the prestress and that compression of MTs balanced a significant, but a relatively small fraction of the prestress. Taken together, these results do not disprove other models of cell deformability, nor they prove tensegrity. However, they do support a priori predictions of tensegrity. As such, it may not be necessary to invoke more complex mechanisms to explain these central features of cell deformability.