Power laws in microrheology experiments on living cells: comparative analysis and modelling

We compare and synthesize the results of two microrheological experiments on the cytoskeleton of single cells. In the first one, the creep function J(t) of a cell stretched between two glass plates is measured after applying a constant force step. In the second one, a micrometric bead specifically bound to transmembrane receptors is driven by an oscillating optical trap, and the viscoelastic coefficient $G_e(\omega)$ is retrieved. Both $J(t)$ and $G_e(\omega)$ exhibit power law behavior: $J(t)= A(t/t_0)^\alpha$ and $\bar G_e(\omega)\bar = G_0 (\omega/\omega_0)^\alpha$, with the same exponent $\alpha\approx 0.2$. This power law behavior is very robust ; $\alpha$ is distributed over a narrow range, and shows almost no dependance on the cell type, on the nature of the protein complex which transmits the mechanical stress, nor on the typical length scale of the experiment. On the contrary, the prefactors $A_0$ and $G_0$appear very sensitive to these parameters. Whereas the exponents $\alpha$ are normally distributed over the cell population, the prefactors $A_0$ and $G_0$ follow a log-normal repartition. These results are compared with other data published in the litterature. We propose a global interpretation, based on a semi-phenomenological model, which involves a broad distribution of relaxation times in the system. The model predicts the power law behavior and the statistical repartition of the mechanical parameters, as experimentally observed for the cells. Moreover, it leads to an estimate of the largest response time in the cytoskeletal network: $\tau_m \approx 1000$ s.Comment: 47 pages, 14 figures // v2: PDF file is now Acrobat Reader 4 (and up) compatible // v3: Minor typos corrected - The presentation of the model have been substantially rewritten (p. 17-18), in order to give more details - Enhanced description of protocols // v4: Minor corrections in the text : the immersion angles are estimated and not measured // v5: Minor typos corrected. Two references were clarifie

Some features of the mechanical response of living cells to large deformations

Pulmonary cells are normally submitted to large deformations which can become critical in the pathological cases where cells are exposed to both chemical and physical aggressive stresses. These injuries affect among others cell membrane, cytoskeleton structure and cellular functions. The response of cells secondary to such large deformations remains to be understood, whereas the limits for cellular elastic behavior are still under debate. In order to explore these features, we have performed AFM tests which consist in indenting living cells with a long tip over almost the entire cell height (~10 μm). Preliminary results show that the cells exhibit a surprisingly high capability of adaptation without breaking. So, we question in the present study the reasons of such cellular adaptability to large deformations and, more particularly, what mechanical properties at different scales are behind such a specific cellular feature. To understand experimental data, we used two different mechanical models of the probe-cell interaction: a granular-like model and a continuous model. The granular-like model, in which local properties are controlled, represents the interdependence between functional sub-cellular structures and aims at considering the spatial reorganization of intracellular force distribution within the cytoskeleton. The continuous model, in which global properties are controlled, will be used as a basis for comparison since it aims to describe stress distributions assuming a homogeneous cytoplasm. We expect, with this comparative approach, being able to relate global to local mechanical properties and their range of validity before the apparition of cell injury

Some features of the mechanical response of living cells to large deformations

Pulmonary cells are normally submitted to large deformations which can become critical in the pathological cases where cells are exposed to both chemical and physical aggressive stresses. These injuries affect among others cell membrane, cytoskeleton structure and cellular functions. The response of cells secondary to such large deformations remains to be understood, whereas the limits for cellular elastic behavior are still under debate. In order to explore these features, we have performed AFM tests which consist in indenting living cells with a long tip over almost the entire cell height (~10 μm). Preliminary results show that the cells exhibit a surprisingly high capability of adaptation without breaking. So, we question in the present study the reasons of such cellular adaptability to large deformations and, more particularly, what mechanical properties at different scales are behind such a specific cellular feature. To understand experimental data, we used two different mechanical models of the probe-cell interaction: a granular-like model and a continuous model. The granular-like model, in which local properties are controlled, represents the interdependence between functional sub-cellular structures and aims at considering the spatial reorganization of intracellular force distribution within the cytoskeleton. The continuous model, in which global properties are controlled, will be used as a basis for comparison since it aims to describe stress distributions assuming a homogeneous cytoplasm. We expect, with this comparative approach, being able to relate global to local mechanical properties and their range of validity before the apparition of cell injury

Cell mechanics of alveolar epithelial cells (AECs) and macrophages (AMs).

International audienceCell mechanics provides an integrated view of many biological phenomena which are intimately related to cell structure and function. Because breathing constitutes a sustained motion synonymous with life, pulmonary cells are normally designed to support permanent cyclic stretch without breaking, while receiving mechanical cues from their environment. The authors study the mechanical responses of alveolar cells, namely epithelial cells and macrophages, exposed to well-controlled mechanical stress in order to understand pulmonary cell response and function. They discuss the principle, advantages and limits of a cytoskeleton-specific micromanipulation technique, magnetic bead twisting cytometry, potentially applicable in vivo. They also compare the pertinence of various models (e.g., rheological; power law) used to extract cell mechanical properties and discuss cell stress/strain hardening properties and cell dynamic response in relation to the structural tensegrity model. Overall, alveolar cells provide a pertinent model to study the biological processes governing cellular response to controlled stress or strain

Force distribution on multiple bonds controls the kinetics of adhesion in stretched cells.

International audienceWe show herein how mechanical forces at macro or micro scales may affect the biological response at the nanoscale. The reason resides in the intimate link between chemistry and mechanics at the molecular level. These interactions occur under dynamic conditions such as the shear stress induced by flowing blood or the intracellular tension. Thus, resisting removal by mechanical forces, e.g., shear stresses, is a general property of cells provided by cellular adhesion. Using classical models issued from theoretical physics, we review the force regulation phenomena of the single bond. However, to understand the force regulation of cellular adhesion sites, we need to consider the collective behavior of receptor-ligand bonds. We discuss the applicability of single bond theories to describe collective bond behavior. Depending on bond configuration, e.g., presently "parallel" and "zipper", the number of bonds and dissociation forces variably affect the kinetics of multiple bonds. We reveal a marked efficiency of the collective organization to stabilize multiple bonds by sharply increasing bond lifetime compared to single bond. These theoretical predictions are then compared to experimental results of the literature concerning the kinetic parameters of bonds measured by atomic force microscopy and by shear flow. These comparisons reveal that the force-control of bonds strongly depends on whether the force distribution on multiple bonds is homogeneous, e.g., in AFM experiments, or heterogeneous, e.g., in shear flow experiments. This reinforces the need of calculating the stress/strain fields exerted on living tissues or cells at various scales and certainly down to the molecular scale

Mechanical and structural assessment of cortical and deep cytoskeleton reveals substrate-dependent alveolar macrophage remodeling.

International audienceThe sensitivity of alveolar macrophages to substrate properties has been described in a recent paper (F?ol et al., Cell Motil. Cytoskel. 63 (2006), 321-340). It is presently re-analyzed in terms of F-actin structure (assessed from 3D-reconstructions in fixed cells) and mechanical properties (assessed by Magnetic Twisting Cytometry experiments in living cells) of cortical and deep cytoskeleton structures for rigid plastic (Young Modulus: 3 MPa) or glass (70 MPa) substrates and a soft (approximately 0.1 kPa) confluent monolayer of alveolar epithelial cells. The cortical cytoskeleton component (lowest F-actin density) is represented by the rapid and softer viscoelastic compartment while the deep cytoskeleton component (intermediate F-actin density) is represented by the slow and stiffer compartment. Stiffness of both cortical and deep cytoskeleton is significantly decreased when soft confluent monolayer of alveolar epithelial cells replace the rigid plastic substrate while F-actin reconstructions reveal a consistent actin cytoskeleton remodeling observable on both cytoskeleton components

Analysis of nonlinear responses of adherent epithelial cells probed by magnetic bead twisting: A finite element model based on a homogenization approach.

International audienceAn 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

Analysis of nonlinear responses of adherent epithelial cells probed by magnetic bead twisting: A finite element model based on a homogenization approach.

International audienceAn 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

Sensitivity of alveolar macrophages to substrate mechanical and adhesive properties.

In order to understand the sensitivity of alveolar macrophages (AMs) to substrate properties, we have developed a new model of macrophages cultured on substrates of increasing Young's modulus: (i) a monolayer of alveolar epithelial cells representing the supple (approximately 0.1 kPa) physiological substrate, (ii) polyacrylamide gels with two concentrations of bis-acrylamide representing low and high intermediate stiffness (respectively 40 kPa and 160 kPa) and, (iii) a highly rigid surface of plastic or glass (respectively 3 MPa and 70 MPa), the two latter being or not functionalized with type I-collagen. The macrophage response was studied through their shape (characterized by 3D-reconstructions of F-actin structure) and their cytoskeletal stiffness (estimated by transient twisting of magnetic RGD-coated beads and corrected for actual bead immersion). Macrophage shape dramatically changed from rounded to flattened as substrate stiffness increased from soft ((i) and (ii)) to rigid (iii) substrates, indicating a net sensitivity of alveolar macrophages to substrate stiffness but without generating F-actin stress fibers. Macrophage stiffness was also increased by large substrate stiffness increase but this increase was not due to an increase in internal tension assessed by the negligible effect of a F-actin depolymerizing drug (cytochalasine D) on bead twisting. The mechanical sensitivity of AMs could be partly explained by an idealized numerical model describing how low cell height enhances the substrate-stiffness-dependence of the apparent (measured) AM stiffness. Altogether, these results suggest that macrophages are able to probe their physical environment but the mechanosensitive mechanism behind appears quite different from tissue cells, since it occurs at no significant cell-scale prestress, shape changes through minimal actin remodeling and finally an AMs stiffness not affected by the loss in F-actin integrity

Nanoscale characterization of chitosan hydrogels mechanical properties by AFM

International audienc