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Non linear cell mechanics

Ex-vivo measure of stress-strain relationships in populations of living adherent cells by means of ligand-coated ferromagnetic microbeads (mean diameter: 4.5 µm) attached to the transmembrane mechanoreceptors which are linked to the cytoskeleton (CSK), reveal non linear cell mechanical behavior. However, this non linear cell mechanical behaviour is subjected to controversy for various reasons. First, it has not been systematically found. Results seem to depend on the micromanipulation method used and/or the cell type. Second, there is a debate concerning the origin of this stress/strain dependent cell behaviour: it might be related to the CSK structural properties, material properties of its elements, distribution of heterogeneous CSK elements or biological factors such as CSK/ adhesion remodeling. Applying similar magnetic torques may lead to opposed conclusions in terms of stress/strain dependence of mechanical properties because experimental cell conditions are not necessarily the same. Most particularly, small or large cellular deformation may be generated and/or measured, depending on the micromanipulation method used. Bead rotation is standardly below 6°-7° for optical magnetic twisting cytometry (OMTC) (Fabry et al., Phys. Rev. E, 68: 041914, 2003) but it is in the range 15°-60° for the Magnetic Twisting Cytometry (MTC) invented by N. Wang, J. Butler and D. Ingber (Science, 260: 1124-1127, 1993). Using an MTC method similar to the one earlier described by Wang et al., we found in adherent alveolar epithelial cells and for magnetic torques in the range 400-1200 pN.µm, a systematic stress (or strain)-dependent increase in elastic modulus, earlier found for endothelial cells and other tissue cells with the same technique. We have tested different models (30-elements tensegrity structure, continuous medium with elastic material or non linear Yeoh strain energy function, viscoelastic solid model with heterogeneous cortical and deep CSK elements) to fit experimental data obtained with the MTC technique. It appears that in the large range of cell deformations tested during MTC experiments, structural properties, non linear properties of CSK, spatial heterogeneity of CSK properties/substructures, could all reasonably play a role in the non linear stress-strain cell behavior, but the specific contribution of these different factors remains to be determined.


Some times ago, Pr Alexander A. Spector posted an interesting and motivating reply to my preliminary comments on the non linearity of the stress – strain relationship observed while measuring elasticity modulus of adherent epithelial cells in culture by magnetic bead twisting twisting cytometry. We agree with him that understanding living cell behaviour is a quite complicated question which requires considering a number of geometrical, mechanical, biological parameters. These numerous parameters are still quite difficult to control altogether especially in living cells. The cellular models we developed concern large cellular deformation conditions in order to fit the experimental conditions and results we obtained with magnetic twisting cytometry. This is the case for the 30-element tensegrity structure used to study the structural cellular properties or the continuous medium of bead-cell interaction developed on the basis of an hyperelastic cellular material.

From a mechanical point of view, our contribution has been to identify and quantify the role of few geometrical/mechanical parameters in the conditions of large deformations and in relation with a partially homogenized view of the actin structure which connects the bead. The parameters studied by my group are notably:

- angle of bead immersion in the cytoplasm (Laurent V.M. et al., J. Biomech. Eng. 2002, 124: 408-421),

- distance between the bottom of the bead and the substrate studied with the assumption of an hyperelastic cellular medium (Ohayon et al. J. Biomech. Eng., 2004, 126: 685-698). For linear elastic cellular medium see also (Ohayon and Tracqui, 2005, Ann Biomed Eng 33:131-141),

- contribution of dense actin (deep cytoskeleton) and less dense actin (cortical cytoskeleton) to the cytoskeleton mechanical properties measured by bead twisting (see laurent.fodil.annals.pdf),

- contribution of stiffness properties of substrate on bead rotation (see Fereol.pdf)

- contribution of internal tension or its non dimensional expression (the initial strain) to global elastic modulus in the 30-element tensegrity structure (SW_EPJAP.pdf),

- The strain-dependence of elastic and shear modulus of the 30-element tensegrity structure which reveals strain-hardening during extension, strain-softening during compression and strain-softening up to 50%-strain and strain-hardening above (Wendling_JTB_1999),

- oscillatory character of the load applied to the cell based on oscillatory tensegrity structure (see canadas.pdf).

In these models designed to study large cellular deformations locally induced by large bead rotation, some other parameter remain to be studied such as the contribution of protein-ligand sliding or the contribution of unfolding transmembrane proteins which most likely depend on the type of receptor-ligand linkage chosen to attach the bead to the actin cytoskeleton. Other questions remain concerning the stress/strain-associated remodelling occurring during bead twisting and this is why we use to applied magnetic torques for almost identical duration of application (namely about 1 min).

Noteworthy, the non linear behaviour of living cells has been found or at least is suggested by experiments conducted with other micromanipulation methods permitting to apply large cellular deformation. A good example is given by Atomic Force Microscopy experiments performed by Sato et al. (J. Biomech., 2000, 33: 127-135). The non linear strain-stiffening behaviour of biopolymer gels could also contribute to non linear behaviour of the cellular response (Janmey et al., Nature Materials, 2007, 6, 48-51).

The strain-hardening behaviour of living cells still remains to be more deeply investigated both experimentally and theoretically.

In Féréol et al. (Biophys. J., 2009, 96: 2009-2022),
we propose a coupled theoretical and experimental study in order to understand the
mechanism of cell sensitivity to substrate stiffness. To do so, we first consider
that adhesion sites pass through different stages of development, e.g., Initial
Adhesion (IA), Focal Complex (FC), Focal Adhesions (FA), characterized by the
recruitment of an increasing number of constituent components resulting in molecular
reinforcement of the links between CSK and extracellular environment. One
assumption is that as adhesion sites gain in molecular complexity and strength
(without necessarily increasing their area), they lose their dynamic character
and become more stationary, providing an evolutionary cell signaling which contributes
to cell adaptation. First, Newton’s action-reaction principle which governs the
static force equilibrium at a given stationary adhesion site demonstrates that stationary
adhesion sites (FA) could not exhibit such a substrate stiffness sensitivity. Second,
considering that the proper of dynamic adhesion site is to move relatively to
actin filament bundle, it appears that mechanical relaxation of the
extracellular environment as well as intracellular tensional properties tend to
slow down dramatically the “instantaneous” biochemical process of
receptor-ligand binding. Such an approach enlightens that force regulation of dynamic
adhesion site depends on mechanical properties of substrate and intracellular
properties which together act as a loading rate at the onset of a time-dependent
maturation in response to acto-myosin traction force. Early experiments with
optical tweezers used as a calibrated spring have already show that nascent (also
dynamic) adhesion sites match the force they exert on substrates of different stiffness
(Choquet et al. 1999 Cell 88: 39-48). Thus, different cells would produce
different cell responses that adapt to the wide variety of extracellular
mechanical environments and intracellular tensional conditions. We used two
cellular models, i.e., alveolar epithelial cells (AECs) and alveolar macrophages
(AMs), exhibiting markedly different mechanical behaviors (Féréol et al.,
Respir. Physiol. Neurobiol. 2008, 163: 3-16) and adhesion sites respectively in
stationary state (FA) and in dynamic state (podosome type adhesion system: PTA).
The cell sensitivity to substrate stiffness of these two cellular models appears
in good agreement with theoretical predictions.


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