Frequency-Dependent Focal Adhesion Instability and Cell Reorientation Under Cyclic Substrate Stretching

Cell orientation under stretch: A review of experimental findings and mathematical modelling

The key role of electro-chemical signals in cellular processes had been known for many years, but more recently the interplay with mechanics has been put in evidence and attracted substantial research interests. Indeed, the sensitivity of cells to mechanical stimuli coming from the microenvironment turns out to be relevant in many biological and physiological circumstances. In particular, experimental evidence demonstrated that cells on elastic planar substrates undergoing periodic stretches, mimicking native cyclic strains in the tissue where they reside, actively reorient their cytoskeletal stress fibres. At the end of the realignment process, the cell axis forms a certain angle with the main stretching direction. Due to the importance of a deeper understanding of mechanotransduction, such a phenomenon was studied both from the experimental and the mathematical modelling point of view. The aim of this review is to collect and discuss both the experimental results on cell reorientation and the fundamental features of the mathematical models that have been proposed in the literature.

Cellular mechanics of wound formation in single cell layer under cyclic stretching

Wounds can be produced when cells and tissues are subjected to excessive forces, for instance, under pathological conditions or nonphysiological loading. However, the cellular behaviors in the wound formation process are not clear. Here we tested the behaviors of wound formation in the epithelial layer with an in-suit uniaxial stretching device. We found that the wound often nucleates at the position where the cells are dividing. The polarization direction of cells near the wound is preferentially along the wound edge, whereas the cells far from the wound are preferentially perpendicular to the stretching direction. The larger the wound area is, the higher is the aspect ratio of the cells around the wound. Increasing the cell density will strengthen the cell layer. The higher the cell density is, the smaller is the area of the wounds, and the weaker is the effect of stretching on the polarization of the cells. Furthermore, we built a coarse-grained cell model that can explicitly consider the elasticity and viscoelasticity of cells, cell-cell interaction, and cell active stress, by which we simulated the wound formation process and quantitatively analyzed the force and stress fields in the cell layer, particularly around the wound. These analyses reveal the cellular mechanisms of wound formation behaviors in the cell layer under stretching and shed useful light on tissue engineering and regenerative medicine for biomedical applications.

Engineering Biomaterials and Approaches for Mechanical Stretching of Cells in Three Dimensions

Mechanical stretch is widely experienced by cells of different tissues in the human body and plays critical roles in regulating their behaviors. Numerous studies have been devoted to investigating the responses of cells to mechanical stretch, providing us with fruitful findings. However, these findings have been mostly observed from two-dimensional studies and increasing evidence suggests that cells in three dimensions may behave more closely to their in vivo behaviors. While significant efforts and progresses have been made in the engineering of biomaterials and approaches for mechanical stretching of cells in three dimensions, much work remains to be done. Here, we briefly review the state-of-the-art researches in this area, with focus on discussing biomaterial considerations and stretching approaches. We envision that with the development of advanced biomaterials, actuators and microengineering technologies, more versatile and predictive three-dimensional cell stretching models would be available soon for extensive applications in such fields as mechanobiology, tissue engineering, and drug screening.

A theoretical model of collective cell polarization and alignment

Collective cell polarization and alignment play important roles in tissue morphogenesis, wound healing and cancer metastasis. How cells sense the direction and position in these processes, however, has not been fully understood. Here we construct a theoretical model based on describing cell layer as a nemato-elastic medium, by which the cell polarization, cell alignment and cell active contraction are explicitly expressed as functions of components of the nematic order parameter. To determine the order parameter we derive two sets of governing equations, one for the force equilibrium of the system, and the other for the minimization of the system's free energy including the energy of cell polarization and alignment. By solving these coupled governing equations, we can predict the effects of substrate stiffness, geometries of cell layers, external forces and myosin activity on the direction- and position-dependent cell aspect ratio and cell orientation. Moreover, the axisymmetric problem with cells on a ring-like pattern is solved analytically, and the analytical solution for cell aspect ratio are governed by parameter groups which include the stiffness of the cell and the substrate, the strength of myosin activity and the external forces. Our predictions of the cell aspect ratio and orientation are generally comparable to experimental observations. These results show that the pattern of cell polarization is determined by the anisotropic degree of active contractile stress, and suggest a stress-driven polarization mechanism that enables cells to sense their spatial positions to develop direction- and position-dependent behavior. This, in turn, sheds light on the ways to control pattern formation in tissue engineering for potential biomedical applications.

Modeling the mechanosensitivity of fast-crawling cells on cyclically stretched substrates

The mechanosensitivity of cells, which determines how they are able to respond to mechanical signals, is crucial for the functioning of biological systems. Experimentally, this is investigated by studying the reorientation of cells on cyclically stretched substrates. The reorientation depends on the type of cell and on the stretching protocol, but the mechanisms responsible for the response are still not completely understood. Here, we introduce a computational model for fast crawling cells on cyclically stretched substrates that accounts for the sub-cellular elements responsible for cell shape and motility. This includes the dynamics of the cell membrane, the actin cytoskeleton, and the focal adhesions with the stretching substrate. These processes evolve over characteristic time scales that can vary by orders of magnitude and naturally give rise to the frequency dependent reorientation observed experimentally. Depending on which processes are being probed by the stretching and on the type of coupling with the substrate, our simulations predict either no reorientation, a bi-stability in the parallel and perpendicular directions, or a complete reorientation in either the parallel or perpendicular direction. In particular, we show that an asymmetry in the adhesion dynamics during the loading and unloading phases of the stretching, whether it comes from the response of the cell itself or from the precise stretching protocol, can be used to selectively align the cells. Our results provide further evidence for the importance of focal adhesion dynamics in determining the mechanosensitive response of cells, as well as a way to interpret recent experiments.

Collective cell polarization and alignment on curved surfaces

Curvature as an important topological parameter of 3D extra-cellular matrix has drawn growing attention in recent years. But the underlying mechanism that curvature influences cell behaviors has remained unknown. In this study, we seeded cells on semi-cylindrical and hemispheric surfaces and tested cell alignment and polarization. We found that the surface curvature has profound effect on cell behaviors. With the decrease of diameter of the cylinder/sphere (i.e. increase of curvature), the cells would more preferentially align and polarize with large aspect ratio in the axial/peripheral direction. And the behaviors of the alignment and polarization were position-dependent. For example, at the end of the cylinder, the cells preferred to align circumferentially; while in the interior region, the cells preferred to align in the axial direction. We showed that the cell polarization and alignment were closely correlated with the in-plane stresses in cell layer. That is, the cell polarization and alignment were controlled by the maximum shear stress, which drove cells to align and polarize along the maximum principal stress. The curvature could influence the magnitude of the maximum shear stress and thus regulate cell behaviors. This study provided important insights into the mechanisms of surface curvature influencing cell behaviors in tissue morphogenesis. In addition, our theory of the stress dependent cellular polarity provides a generalized interpretation of the curvature and edge effects which might be extended to understand other steric effects in cell behaviors.

A general model of focal adhesion orientation dynamics in response to static and cyclic stretch

Understanding cellular response to mechanical forces is immensely important for a plethora of biological processes. Focal adhesions are multimolecular protein assemblies that connect the cell to the extracellular matrix and play a pivotal role in cell mechanosensing. Under time-varying stretches, focal adhesions dynamically reorganize and reorient and as a result, regulate the response of cells in tissues. Here I present a simple theoretical model based on, to my knowledge, a novel approach in the understanding of stretch-sensitive bond association and dissociation processes together with the elasticity of the cell-substrate system to predict the growth, stability, and the orientation of focal adhesions in the presence of static as well as cyclically varying stretches. The model agrees well with several experimental observations; most importantly, it explains the puzzling observations of parallel orientation of focal adhesions under static stretch and nearly perpendicular orientation in response to fast varying cyclic stretch.

Rumi De presents a model for focal adhesion dynamics under static stretch and cyclic stretch conditions. The predictions agree with prior observations and may explain the puzzling observation that focal adhesions orient toward the parallel direction in the presence of static or quasi-static stretch, but toward the perpendicular direction under fast varying cyclic stretch.

Cellular mechanosensing of the biophysical microenvironment: A review of mathematical models of biophysical regulation of cell responses

Cells in vivo reside within complex microenvironments composed of both biochemical and biophysical cues. The dynamic feedback between cells and their microenvironments hinges upon biophysical cues that regulate critical cellular behaviors. Understanding this regulation from sensing to reaction to feedback is therefore critical, and a large effort is afoot to identify and mathematically model the fundamental mechanobiological mechanisms underlying this regulation. This review provides a critical perspective on recent progress in mathematical models for the responses of cells to the biophysical cues in their microenvironments, including dynamic strain, osmotic shock, fluid shear stress, mechanical force, matrix rigidity, porosity, and matrix shape. The review highlights key successes and failings of existing models, and discusses future opportunities and challenges in the field.

Mechanics of Cell Mechanosensing in Protrusion and Retraction of Lamellipodium

Lamellipodia (LP), a subcellular structure at cell front, plays a key role in cell spreading and migration. And its mechanosensing function is of crucial importance for cell activities. But the mechanism of the mechanosensing function remains poorly understood. Here we developed a multiscale model to consider its protrusion and retraction processes, and analyzed the forces acted on the key structural components of the LP and the effect of these forces on LP movement. Our results show that raising substrate rigidity increases the force acting on the focal adhesion (FA) and decreases the force on LP actin, thus promoting the maturation of FA while suppressing the detachment of LP actin from the cell membrane. The membrane tension also influences the LP movement, but its effect is opposite to that of the substrate rigidity. It turns out that the substrate rigidity and membrane tension together regulate the dynamics of FAs and the detachment of LP actin, which in turn determine the LP movement. Interestingly, we found that the effect of substrate rigidity and membrane tension on the LP movement both exhibit a biphasic manner. We show that our predictions agree, in general, with the experiments on cell mechanosensing behaviors at both subcellular and cellular levels.

Cell Contractility Facilitates Alignment of Cells and Tissues to Static Uniaxial Stretch

During animal development and homeostasis, the structure of tissues, including muscles, blood vessels, and connective tissues, adapts to mechanical strains in the extracellular matrix (ECM). These strains originate from the differential growth of tissues or forces due to muscle contraction or gravity. Here we show using a computational model that by amplifying local strain cues, active cell contractility can facilitate and accelerate the reorientation of single cells to static strains. At the collective cell level, the model simulations show that active cell contractility can facilitate the formation of strings along the orientation of stretch. The computational model is based on a hybrid cellular Potts and finite-element simulation framework describing a mechanical cell-substrate feedback, where: 1) cells apply forces on the ECM, such that 2) local strains are generated in the ECM and 3) cells preferentially extend protrusions along the strain orientation. In accordance with experimental observations, simulated cells align and form stringlike structures parallel to static uniaxial stretch. Our model simulations predict that the magnitude of the uniaxial stretch and the strength of the contractile forces regulate a gradual transition between stringlike patterns and vascular networklike patterns. Our simulations also suggest that at high population densities, less cell cohesion promotes string formation.

Dissecting Collective Cell Behavior in Polarization and Alignment on Micropatterned Substrates

Pattern-dependent collective behaviors of cells have recently raised intensive attention. However, the underlying mechanisms that regulate these behaviors are largely elusive. Here, we report a quantitative study, combining experiment and modeling, on cell polarization and arrangement on a micropatterned substrate. We show that cells exhibit position-dependent collective behaviors that can be regulated by geometry and stiffness of the patterned substrate. We find that the driving force for these collective behaviors is the in-plane maximum shear stress in the cell layer that directs the arrangement of cells. The larger the shear stress, the more the cells preferentially align and polarize along the direction of the maximum principal stress. We also find that the aspect ratio of cell polarization shape and the degree to which cells preferentially align along the direction of maximum principal stress exhibit a biphasic dependence on substrate rigidity, corresponding to our quantitative predictions that the magnitude of the maximum shear stress is biphasically dependent on the stiffness of the substrate. As such, the driving force of these cell collective behaviors can be quantified using the maximum shear stress.

Modeling Active Mechanosensing in Cell-Matrix Interactions

Cells actively sense the mechanical properties of the extracellular matrix, such as its rigidity, morphology, and deformation. The cell-matrix interaction influences a range of cellular processes, including cell adhesion, migration, and differentiation, among others. This article aims to review some of the recent progress that has been made in modeling mechanosensing in cell-matrix interactions at different length scales. The issues discussed include specific interactions between proteins, the structure and mechanosensitivity of focal adhesions, the cluster effects of the specific binding, the structure and behavior of stress fibers, cells' sensing of substrate stiffness, and cell reorientation on cyclically stretched substrates. The review concludes by looking toward future opportunities in the field and at the challenges to understanding active cell-matrix interactions.

The many ways adherent cells respond to applied stretch

Cells in various tissues are subjected to mechanical stress and strain that have profound effects on cell architecture and function. The specific response of the cell to applied strain depends on multiple factors, including cell contractility, spatial and temporal strain pattern, and substrate dimensionality and rigidity. Recent work has demonstrated that the cell response to applied strain depends on a complex combination of these factors, but the way these factors interact to elicit a specific response is not intuitive. We submit that an understanding of the integrated response of a cell to these factors will provide new insight into mechanobiology and contribute to the effective design of deformable engineered scaffolds meant to provide appropriate mechanical cues to the resident cells.

Shear-induced force transmission in a multicomponent, multicell model of the endothelium

Haemodynamic forces applied at the apical surface of vascular endothelial cells (ECs) provide the mechanical signals at intracellular organelles and through the inter-connected cellular network. The objective of this study is to quantify the intracellular and intercellular stresses in a confluent vascular EC monolayer. A novel three-dimensional, multiscale and multicomponent model of focally adhered ECs is developed to account for the role of potential mechanosensors (glycocalyx layer, actin cortical layer, nucleus, cytoskeleton, focal adhesions (FAs) and adherens junctions (ADJs)) in mechanotransmission and EC deformation. The overriding issue addressed is the stress amplification in these regions, which may play a role in subcellular localization of mechanotransmission. The model predicts that the stresses are amplified 250-600-fold over apical values at ADJs and 175-200-fold at FAs for ECs exposed to a mean shear stress of 10 dyne cm(-2). Estimates of forces per molecule in the cell attachment points to the external cellular matrix and cell-cell adhesion points are of the order of 8 pN at FAs and as high as 3 pN at ADJs, suggesting that direct force-induced mechanotransmission by single molecules is possible in both. The maximum deformation of an EC in the monolayer is calculated as 400 nm in response to a mean shear stress of 1 Pa applied over the EC surface which is in accord with measurements. The model also predicts that the magnitude of the cell-cell junction inclination angle is independent of the cytoskeleton and glycocalyx. The inclination angle of the cell-cell junction is calculated to be 6.6° in an EC monolayer, which is somewhat below the measured value (9.9°) reported previously for ECs subjected to 1.6 Pa shear stress for 30 min. The present model is able, for the first time, to cross the boundaries between different length scales in order to provide a global view of potential locations of mechanotransmission.

A mechanochemical model of cell reorientation on substrates under cyclic stretch

We report a theoretical study on the cyclic stretch-induced reorientation of spindle-shaped cells. Specifically, by taking into account the evolution of sub-cellular structures like the contractile stress fibers and adhesive receptor-ligand clusters, we develop a mechanochemical model to describe the dynamics of cell realignment in response to cyclically stretched substrates. Our main hypothesis is that cells tend to orient in the direction where the formation of stress fibers is energetically most favorable. We show that, when subjected to cyclic stretch, the final alignment of cells reflects the competition between the elevated force within stress fibers that accelerates their disassembly and the disruption of cell-substrate adhesion as well, and an effectively increased substrate rigidity that promotes more stable focal adhesions. Our model predictions are consistent with various observations like the substrate rigidity dependent formation of stable adhesions and the stretching frequency, as well as stretching amplitude, dependence of cell realignment. This theory also provides a simple explanation on the regulation of protein Rho in the formation of stretch-induced stress fibers in cells.

Impact of cell shape on cell migration behavior on elastic substrate

Cell shape is known to have profound effects on a number of cell behaviors. In this paper we have studied its role in cell migration through modeling the effect of cell shape on the cell traction force distribution, the traction force dependent stability of cell adhesion and the matrix rigidity dependent traction force formation. To quantify the driving force of cell migration, a new parameter called the motility factor, that takes account of the effect of cell shape, matrix rigidity and dynamic stability of cell adhesion, is proposed. We showed that the motility factor depends on the matrix rigidity in a biphasic manner, which is consistent with the experimental observations of the biphasic dependence of cell migration speed on the matrix rigidity. We showed that the cell shape plays a pivotal role in the cell migration behavior by regulating the traction force at the cell front and rear. The larger the cell polarity, the larger the motility factor is. The keratocyte-like shape has a larger motility factor than the fibroblast-like shape, which explains why keratocyte has a much higher migration speed. The motility factor might be an appropriate parameter for a quantitative description of the driving force of cell migration.

Cell adhesion nucleation regulated by substrate stiffness: a Monte Carlo study

Cell adhesions are modulated by the interactions between cells and their surroundings, among which substrate stiffness plays an important role in mediating cellular behaviors and functions. Little is known, however, about the inherent mechanism of how nascent adhesion nucleation, as the precursor of focal adhesions, is regulated by substrate stiffness. This paper presents a microscopic model to imitate integrin clustering kinetics, where integrin diffusion, activation on elastic substrates, receptor-ligand binding and association dynamics are fully considered. Particularly, the contribution of substrate compliance to the activation energy is analyzed, leading to a description of mechanical energy barrier for stretching a substrate-bound integrin molecule from bent to extended conformations. A series of Monte Carlo simulations for integrin clustering dynamics are performed with varied substrate Young's moduli, which demonstrates that more integrins are clustered on stiffer substrates once they begin to assemble over a rigidity threshold, indicating the responsiveness of adhesion nucleation to substrate elasticity, which is in reasonable agreement with results reported previously. Also, these simulations show that the sensitivity of integrin clustering to substrate stiffness is mediated by chemical affinity between receptor-ligand pairs and that between integrins cross-linked by adapter proteins, as well as integrin density on cell membranes. The investigation offers a fascinating insight into the inherent mechanism of mechanosensing concerning integrin-mediated cell-matrix initial adhesions.