Original Research Article
Cell orientation under stretch: Stability of a linear viscoelastic model

https://doi.org/10.1016/j.mbs.2021.108630Get rights and content

Highlights

  • Under a cyclic strain, cells orient their stress fibres in a precise direction.

  • The model describes the cell reorientation dynamics using a viscoelastic approach.

  • The equilibrium orientation is predicted through an orthotropic energy minimization.

  • Viscous effects slow down the reorientation of the cell towards the predicted angle.

  • Simulations show a viscous–elastic transition as the oscillation frequency varies.

Abstract

The sensitivity of cells to alterations in the microenvironment and in particular to external mechanical stimuli is significant in many biological and physiological circumstances. In this regard, experimental assays demonstrated that, when a monolayer of cells cultured on an elastic substrate is subject to an external cyclic stretch with a sufficiently high frequency, a reorganization of actin stress fibres and focal adhesions happens in order to reach a stable equilibrium orientation, characterized by a precise angle between the cell major axis and the largest strain direction. To examine the frequency effect on the orientation dynamics, we propose a linear viscoelastic model that describes the coupled evolution of the cellular stress and the orientation angle. We find that cell orientation oscillates tending to an angle that is predicted by the minimization of a very general orthotropic elastic energy, as confirmed by a bifurcation analysis. Moreover, simulations show that the speed of convergence towards the predicted equilibrium orientation presents a changeover related to the viscous–elastic transition for viscoelastic materials. In particular, when the imposed oscillation period is lower than the characteristic turnover rate of the cytoskeleton and of adhesion molecules such as integrins, reorientation is significantly faster.

Introduction

During their life cycle, cells are constantly exposed to numerous stimuli coming from the surrounding microenvironment. The nature of these cues is wide-ranging: among them, a significant role is played by mechanical prompts, since many experiments demonstrated that they trigger a cellular response [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. Cell sensitivity to mechanical actions is relevant in many biological and physiological circumstances, such as growth, differentiation, motility, apoptosis and tissue fibrosis [6], [9], [11]. Its precise understanding has then gathered increasing research attention, since it could be helpful to acquire a deeper knowledge of some diseases and of morphogenesis, just to mention a few examples. For instance, an altered perception of mechanical stimuli due to cell–cell contact inhibition and to cell tensile stress is known to have a role in tumourigenesis and cancer development [12], [13], [14], [15], and it is also related to epithelial–mesenchymal transition [16], [17] in neoplastic tissues.

Moreover, the active response of the cell to mechanical interactions with the environment is involved in cell culturing, development, and tissue engineering. In particular, during embryogenesis, the formation of residual stresses and active forces is believed to drive heart formation and looping [18]. Cardiac cell cultures also display enhanced hypertrophy, proliferation and alignment when subject to static or cyclic stress [19]. Notably, experimental tests carried out on several types of cells (like fibroblasts, myofibroblasts, cardiomyocytes and endothelial cells) showed that alignment in response to a deformation is a common feature which proves their capability to adapt after mechanical stimuli [1], [3], [4], [8], [20], [21], [22]. In detail, when a monolayer undergoes a cyclic deformation, cells lying on the substrate tend to change their orientation in a precise way, until they reach a stable configuration characterized by a well-defined angle between their major axis and the direction of largest stretching. In this process, a fundamental role is played by the cytoskeleton [8], [23], [24], [25]: focal adhesions (FAs), i.e. protein complexes which provide cell contact with the substrate and the extracellular matrix, sense the mechanical stress and induce a remodelling of the cytoskeletal structure, through the formation of oriented actin stress fibres (SFs). These fibre bundles are able to develop contractile forces: when submitted to an external stretch, the cell reorganizes its SF structure, disrupting and rebuilding them in a specific direction to relieve the stress.

Several works, both on the experimental [1], [4], [7], [8], [20], [23], [26], [27], [28], [29], [30] and on the theoretical side [3], [25], [31], [32], [33], [34], [35], [36], have tried to address the problem of cell reorientation under mechanical stretch. It is recognized that this mechanism is actively performed by the cell [37], and that it is induced by mechanical strain deforming the substrate to which they adhere. Moreover, there is common agreement on the fact that cells on a plane substrate undergoing a cyclic deformation orient their stress fibres in a direction which is oblique or in some cases perpendicular to the applied strain [3], [20], [23], [38]. Indeed, when the substrate deformation is transmitted to the cell cytoskeleton through FAs, a reorganization of SF structure happens: they are disassembled and rebuilt in a precise direction [39], fostering changes in shape and orientation of the whole cell. In addition, FAs themselves form clusters at the ends of aligned SFs, giving the cell an elongated and clearly oriented morphology (see Fig. 1). Therefore it is possible to define an equilibrium orientation angle, θeq, that is the angle formed by the cell major axis and the direction of stretching when cell orientation does not evolve anymore. In this respect, mathematical models trying to predict this equilibrium orientation angle and the driving force of such a behaviour have been proposed, using different approaches but mainly in a linear elastic framework. For instance, the first attempts to describe cell orientation suggested a preference for the minimal strain or minimal stress directions [20], [24], [27], [29], [33], [34]. Looking closely at biaxial tests, Livne and coworkers [20] found a linear relationship between cos2θeq and a parameter quantifying the biaxiality of the deformation.

Remarkably, some experimental assays were performed applying deformations for which using linear elasticity should be theoretically inaccurate (e.g., up to 24% in [20] and up to 32% in [27]). Starting from this experimental evidence, Lucci and Preziosi [40] proved that a generalization of the linear relationship found in the linear elastic case by Livne et al. [20] also holds for a very large class of nonlinear constitutive orthotropic models. In the nonlinear framework, the squared cosine of the orientation angle is linearly dependent on a parameter which is the natural generalization of the one found in [20], with a slope depending on a combination of elastic coefficients characterizing the nonlinear strain energy.

Nevertheless, there are other factors influencing the orientation of the cell, the most relevant of which is probably the frequency of the applied cyclic deformation [28], [41]. In fact, it has been observed that, in order to trigger such a response, the period of the stretching cycle must be sufficiently small [28], [42]. As specified in [28], this threshold seems to be cell-type dependent, leading to minimum frequencies that go from 0.01Hz for rat embryonic fibroblasts to 0.1Hz for human dermal fibroblasts. This mechanism cannot be covered by the purely elastic descriptions discussed above, and calls for the introduction of a characteristic response time that needs to be compared with the periodic deformation time scale. The existence of such a characteristic time might be related to the reorganization of the acto-myosin cytoskeleton and of the ensemble of focal adhesions with the substrate. Indeed, it is known that the characteristic turnover times of both phenomena are of the order of tens of seconds, or even minutes (see, for instance, [26], [43], [44], [45]).

On the basis of this observation, in this paper we propose a viscoelastic model for cell preferential orientations, in order to describe reorganization processes occurring inside the cell and between the cell and its microenvironment when a mechanical deformation is applied to the substrate. To our knowledge, previous viscoelastic descriptions of cell stress fibre dynamics have been mainly focused on the microscopic scale [46], [47], while in this article we treat the monolayer as a continuum. In particular, we introduce an anisotropic viscoelastic description that couples the evolution of the SF orientation angle with the mechanical stress exerted on the cell as a consequence of cyclic stretching. Hence, the ensemble of cells lying on the substrate is considered as a Maxwell orthotropic fluid with a single relaxation time. We prove that, for high stretching frequencies, the cell cytoskeleton does not have enough time to reorganize and behaves elastically, while for slow processes the viscous character emerges.

Furthermore, after having showed that the steady angles are predicted by an energy minimization, we work with a very general orthotropic material. An extensive bifurcation analysis is then performed, discussing the role of elastic parameters and finding the conditions under which a certain angle of cell orientation is stable. We find that also in this general set up there exists a linear relationship between cos2θeq and a combination of parameters of the orthotropic elasticity tensor. The slope of the straight line fitting experimental data suggests that, among all coefficients, a more relevant role is played by those in charge of describing the cell response to elongation along its orientation axis and to shear.

Finally, we perform some numerical simulations using the complete viscoelastic model, studying the reorientation dynamics in the high frequency and low frequency cases together with stress evolution. It is found that the cell orientation angle evolves towards the steady state predicted by the linear stability analysis, with a speed which depends on the elastic or viscous character of the system. Moreover, in accordance with the observation in [28], [42], simulations show that the speed of reorientation towards the equilibrium angle sensibly depends on the frequency of imposed oscillations. In particular, it presents a transition for values of the ratio of the oscillation period and the characteristic time of viscoelasticity close to 2π, so that the time required to observe reorientation is of the order of days for smaller frequencies, saturating to one hour for larger frequencies.

In detail, the paper is organized as follows. In Section 2 the general mathematical model is introduced, deriving the equations for the viscoelastic system and studying their significant limits, i.e. the high-frequency and low-frequency cases. Section 3 is devoted to a detailed bifurcation analysis of the model for an orthotropic energy density, deriving conditions under which equilibrium orientations are stable. In Section 4 we discuss the implementation and report some numerical results of our model, showing both the elastic and the viscous behaviour of the system. Finally, Section 5 is dedicated to a summary of the results and to the discussion of some open issues, which may be of interest for future research. InAppendix we report some details related to the possible presence of a symmetry breaking phenomenon.

Section snippets

Viscoelastic model

We consider a two-dimensional substrate seeded of cells that is stretched biaxially. While the response of the extracellular material is in general isotropic and elastic, the mechanical behaviour of the ensemble of cells can be regarded as anisotropic and viscoelastic. The viscoelastic character is due to the reorganization of the acto-myosin network inside the cell and to the rearrangement of focal adhesions (FAs), performed through repeated detachments and attachments of integrin bonds with

Bifurcation analysis

In this Section, we study equilibrium orientations and their bifurcations. Our goal is to describe the monolayer subject to a periodic stretch through its elastic energy, since the steady orientation of the cells is predicted by its minimization as discussed above. This allows us to study in detail the equilibrium angles in terms of a very general strain energy, looking for those orientations which minimize it for a fixed deformation. Finally, we draw bifurcation diagrams in terms of a

Simulations of the viscoelastic model

After having discussed the equilibrium orientations in Section 3, here we focus on the dynamics of cell reorientation in response to the viscoelastic model presented in Section 2, performing some numerical simulations. More specifically, we consider the system of equations which describes the time evolution of the Cauchy stress tensor T and the reorientation dynamics of the angle θ. As regards the former, its evolution is governed by the viscoelastic constitutive equation (6) described in

Discussion

The response of cells to mechanical cues is a relevant biological phenomenon which still needs investigations and efforts to be enlightened. Starting from experimental observations showing that, when a monolayer is subject to a biaxial stretch, cells orient themselves in a well-defined configuration, in this paper we employed mechanical instruments to further explore this behaviour, focusing on linear elasticity and viscoelasticity. Previous works [20], [40] suggested that a linear elastic

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was partially supported by MIUR (Italian Ministry of Education, Universities and Research) through the PRIN project n. 2017KL4EF3 on “Mathematics of active materials: From mechanobiology to smart devices” and through the “Dipartimento di Eccellenza” 2018–2022 project n. E11G18000350001, and by the National Group of Mathematical Physics (GNFM-INdAM) grant “Progetto Giovani 2020”.

References (68)

  • XuG. et al.

    A tensegrity model of cell reorientation on cyclically stretched substrates

    Biophys. J.

    (2016)
  • LiuB. et al.

    Role of cyclic strain frequency in regulating the alignment of vascular smooth muscle cells in vitro

    Biophys. J.

    (2008)
  • PalamidessiA. et al.

    The GTPase-activating protein RN-3 controls focal adhesion turnover and cell migration

    Curr. Biol.

    (2013)
  • KongD. et al.

    Stability of adhesion clusters and cell reorientation under lateral cyclic tension

    Biophys. J.

    (2008)
  • Civelekoglu-ScholeyG. et al.

    Model of coupled transient changes of Rac, Rho, adhesions and stress fibers alignment in endothelial cells responding to shear stress

    J. Theoret. Biol.

    (2005)
  • LiZ. et al.

    Molecular mechanisms of mechanotransduction in integrin-mediated cell–matrix adhesion

    Exp. Cell Res.

    (2016)
  • ZhuC. et al.

    Dynamic bonds and their roles in mechanosensing

    Curr. Opin. Chem. Biol.

    (2019)
  • TondonA. et al.

    Dependence of cyclic stretch-induced stress fiber reorientation on stretch waveform

    J. Biomech.

    (2012)
  • CiambellaJ. et al.

    A structurally frame-indifferent model for anisotropic visco-hyperelastic materials

    J. Mech. Phys. Solids

    (2021)
  • DeshpandeV. et al.

    Chemo-mechanical model of a cell as a stochastic active gel

    J. Mech. Phys. Solids

    (2021)
  • BuskermolenA.B.C. et al.

    Entropic forces drive cellular contact guidance

    Biophys. J.

    (2019)
  • ButlerJ.P. et al.

    Traction fields, moments, and strain energy that cells exert on their surroundings

    Am. J. Physiol.: Cell Physiol.

    (2002)
  • ChenB. et al.

    Cyclic stretch induces cell reorientation on substrates by destabilizing catch bonds in focal adhesions

    PLoS One

    (2012)
  • CollinsworthA.M. et al.

    Orientation and length of mammalian skeletal myocytes in response to a unidirectional stretch

    Cell Tissue Res.

    (2000)
  • EastwoodM. et al.

    Effect of precise mechanical loading on fibroblast populated collagen lattices: Morphological changes

    Cell. Motil. Cytoskeleton

    (1998)
  • IngberD.

    Mechanobiology and diseases of mechanotransduction

    Ann. Med.

    (2009)
  • KimB.-S. et al.

    Cyclic mechanical strain regulates the development of engineered smooth muscle tissue

    Nat. Biotechnol.

    (1999)
  • YoonJ.-K. et al.

    Stretchable piezoelectric substrate providing pulsatile mechanoelectric cues for cardiomyogenic differentiation of mesenchymal stem cells

    ACS Appl. Mater. Interfaces

    (2017)
  • CarverW. et al.

    Regulation of tissue fibrosis by the biomechanical environment

    Biomed Res. Int.

    (2013)
  • ButcherD.T. et al.

    A tense situation: forcing tumour progression

    Nat. Rev. Cancer

    (2009)
  • ChaplainM. et al.

    Mathematical modelling of the loss of tissue compression responsiveness and its role in solid tumour development

    Math. Med. Biol.

    (2006)
  • KassJ. et al.

    Mammary epithelial cell: Influence of ECM composition and organization during development and tumorigenesis

    Int. J. Biochem. Cell Biol.

    (2007)
  • KumarS. et al.

    Mechanics, malignancy, and metastasis: the force journey of a tumor cell

    Cancer Metastasis Rev.

    (2009)
  • KalluriR. et al.

    The basics of epithelial-mesenchymal transition

    J. Clin. Invest.

    (2009)
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