Models of cytoskeletal mechanics of adherent cells

 Adherent cells sense their mechanical environment, which, in turn, regulates their functions. During the past decade, a growing body of evidence has indicated that a deformable, solid-state intracellular structure known as the cytoskeleton (CSK) plays a major role in transmitting and distributing mechanical stresses within the cell as well as in their conversion into a chemical response. Therefore in order to understand mechanical regulation and control of cellular functions, one needs to understand mechanisms that determine how the CSK changes its shape and mechanics in response to stress. In this survey, we examined commonly used structurally based models of the CSK. In particular, we focused on two classes of these models: open-cell foam networks and stress-supported structures. We identified the underlying mechanisms that determine deformability of those models and compare model predictions with data previously obtained from mechanical tests on cultured living adherent cells at steady state. We concluded that stress-supported structures appear more suitable for describing cell deformability because this class of structures can explain the central role that the cytoskeletal contractile prestress plays in cellular mechanics. Received: 2 January 2002 / Accepted: 27 February 2002     

A phenomenological cohesive model for the macroscopic simulation of cell–matrix adhesions

Cell adhesion is crucial for cells to not only physically interact with each other but also sense their microenvironment and respond accordingly. In fact, adherent cells can generate physical forces that are transmitted to the surrounding matrix, regulating the formation of cell–matrix adhesions. The main purpose of this work is to develop a computational model to simulate the dynamics of cell–matrix adhesions through a cohesive formulation within the framework of the finite element method and based on the principles of continuum damage mechanics. This model enables the simulation of the mechanical adhesion between cell and extracellular matrix (ECM) as regulated by local multidirectional forces and thus predicts the onset and growth of the adhesion. In addition, this numerical approach allows the simulation of the cell as a whole, as it models the complete mechanical interaction between cell and ECM. As a result, we can investigate and quantify how different mechanical conditions in the cell (e.g., contractile forces, actin cytoskeletal properties) or in the ECM (e.g., stiffness, external forces) can regulate the dynamics of cell–matrix adhesions.     

Mechanics of the F-actin cytoskeleton

Dynamic regulation of the filamentous actin (F-actin) cytoskeleton is critical to numerous physical cellular processes, including cell adhesion, migration and division. Each of these processes require precise regulation of cell shape and mechanical force generation which, to a large degree, is regulated by the dynamic mechanical behaviors of a diverse assortment of F-actin networks and bundles. In this review, we review the current understanding of the mechanics of F-actin networks and identify areas of further research needed to establish physical models. We first review our understanding of the mechanical behaviors of F-actin networks reconstituted in vitro, with a focus on the nonlinear mechanical response and behavior of “active” F-actin networks. We then explore the types of mechanical response measured of cytoskeletal F-actin networks and bundles formed in living cells and identify how these measurements correspond to those performed on reconstituted F-actin networks formed in vitro. Together, these approaches identify the challenges and opportunities in the study of living cytoskeletal matter.     

A Prestressed Cable Network Model of the Adherent Cell Cytoskeleton

A prestressed cable network is used to model the deformability of the adherent cell actin cytoskeleton. The overall and microstructural model geometries and cable mechanical properties were assigned values based on observations from living cells and mechanical measurements on isolated actin filaments, respectively. The models were deformed to mimic cell poking (CP), magnetic twisting cytometry (MTC) and magnetic bead microrheometry (MBM) measurements on living adherent cells. The models qualitatively and quantitatively captured the fibroblast cell response to the deformation imposed by CP while exhibiting only some qualitative features of the cell response to MTC and MBM. The model for CP revealed that the tensed peripheral actin filaments provide the key resistance to indentation. The actin filament tension that provides mechanical integrity to the network was estimated at ~158 pN, and the nonlinear mechanical response during CP originates from filament kinematics. The MTC and MBM simulations revealed that the model is incomplete, however, these simulations show cable tension as a key determinant of the model response.     

A cellular tensegrity model to analyse the structural viscoelasticity of the cytoskeleton

This study describes the viscoelastic properties of a refined cellular-tensegrity model composed of six rigid bars connected to a continuous network of 24 viscoelastic pre-stretched cables (Voigt bodies) in order to analyse the role of the cytoskeleton spatial rearrangement on the viscoelastic response of living adherent cells. This structural contribution was determined from the relationships between the global viscoelastic properties of the tensegrity model, i.e., normalized viscosity modulus (eta(*)), normalized elasticity modulus (E(*)), and the physical properties of the constitutive elements, i.e., their normalized length (L(*)) and normalized initial internal tension (T(*)). We used a numerical method to simulate the deformation of the structure in response to different types of loading, while varying by several orders of magnitude L(*) and T(*). The numerical results obtained reveal that eta(*) remains almost independent of changes in T(*) (eta(*) proportional, variant T(*+0.1)), whereas E(*) increases with approximately the square root of the internal tension T(*) (from E(*) proportional, variant T(*+0.3) to E(*) proportional, variant T(*+0.7)). Moreover, structural viscosity eta(*) and elasticity E(*) are both inversely proportional to the square of the size of the structure (eta(*) proportional, variant L(*-2) and E(*) proportional, variant L(*-2)). These structural properties appear consistent with cytoskeleton (CSK) mechanical properties measured experimentally by various methods which are specific to the CSK micromanipulation in living adherent cells. Present results suggest, for the first time, that the effect of structural rearrangement of CSK elements on global CSK behavior is characterized by a faster cellular mechanical response relatively to the CSK element response, which thus contributes to the solidification process observed in adherent cells. In extending to the viscoelastic properties the analysis of the mechanical response of the cellular 30-element tensegrity model, the present study contributes to the understanding of recent results on the cellular-dynamic response and allows to reunify the scattered data reported for the viscoelastic properties of living adherent cells.     

On atomic force microscopy and the constitutive behavior of living cells

Atomic force microscopy (AFM) is one of many new technologies available to study the mechanical properties and mechanobiological responses of living cells. Despite the widespread usage of this technology, there has been little attempt to develop new theoretical frameworks to interpret the associated data. Rather, most analyses rely on the classical Hertz solution for the indentation of an elastic half-space within the context of linearized elasticity. In contrast, we propose a fully nonlinear, constrained mixture model for adherent cells that allows one to account separately for the contributions of the three primary structural constituents of the cytoskeleton. Moreover, we extend a prior solution for a small indentation superimposed on a finite equibiaxial extension by incorporating in this mixture model for the special case of an initially random distribution of constituents (actin, intermediate filaments, and microtubules). We submit that this theoretical framework will allow an improved interpretation of indentation force-depth data from a sub-class of atomic force microscopy tests and will serve as an important analytical check for future finite element models. The latter will be necessary to exploit further the capabilities of both atomic force microscopy and nonlinear mixture theories for cell behavior.     

Cell Mechanics,Structure, and Function Are Regulated by the Stiffness of the Three-Dimensional Microenvironment

This study adopts a combined computational and experimental approach to determine the mechanical, structural, and metabolic properties of isolated chondrocytes cultured within three-dimensional hydrogels. A series of linear elastic and hyperelastic finite-element models demonstrated that chondrocytes cultured for 24 h in gels for which the relaxation modulus is <5 kPa exhibit a cellular Young’s modulus of ～5 kPa. This is notably greater than that reported for isolated chondrocytes in suspension. The increase in cell modulus occurs over a 24-h period and is associated with an increase in the organization of the cortical actin cytoskeleton, which is known to regulate cell mechanics. However, there was a reduction in chromatin condensation, suggesting that changes in the nucleus mechanics may not be involved. Comparison of cells in 1% and 3% agarose showed that cells in the stiffer gels rapidly develop a higher Young’s modulus of ～20 kPa, sixfold greater than that observed in the softer gels. This was associated with higher levels of actin organization and chromatin condensation, but only after 24 h in culture. Further studies revealed that cells in stiffer gels synthesize less extracellular matrix over a 28-day culture period. Hence, this study demonstrates that the properties of the three-dimensional microenvironment regulate the mechanical, structural, and metabolic properties of living cells.     

The stress and strain of cytokinesis

The ultimate goal of all signaling pathways in cytokinesis is to control the mechanical separation of the mother cell into two daughter cells. Because of the intrinsic mechanical nature of cytokinesis, it is essential to understand fully how cell shapes and the material properties of the cell are generated, how these shapes and material properties create force, and how motor proteins such as myosin-II modify the system to achieve successful cytokinesis. In this review (which is part of the Cytokinesis series), we discuss the relevant physical properties of cells, how these properties are measured and the basic models that are used to understand cell mechanics. Finally, we present our current understanding of how cytokinesis mechanics work.     

Imaging and manipulating the structural machinery of living cells on the micro- and nanoscale

The structure, physiology, and fate of living cells are all highly sensitive to mechanical forces in the cellular microenvironment, including stresses and strains that originate from encounters with the extracellular matrix (ECM), blood and other flowing materials, and neighbouring cells. This relationship between context and physiology bears tremendous implications for the design of cellular micro-or nanotechnologies, since any attempt to control cell behavior in a device must provide the appropriate physical microenvironment for the desired cell behavior. Cells sense, process, and respond to biophysical cues in their environment through a set of integrated, multi-scale structural complexes that span length scales from single molecules to tens of microns, including small clusters of force-sensing molecules at the cell surface, micron-sized cell-ECM focal adhesion complexes, and the cytoskeleton that permeates and defines the entire cell. This review focuses on several key technologies that have recently been developed or adapted for the study of the dynamics of structural micro-and nanosystems in living cells and how these systems contribute to spatially-and temporally-controlled changes in cellular structure and mechanics. We begin by discussing subcellular laser ablation, which permits the precise incision of nanoscale structural elements in living cells in order to discern their mechanical properties and contributions to cell structure. We then discuss fluorescence recovery after photobleaching and fluorescent speckle microscopy, two live-cell fluorescence imaging methods that enable quantitative measurement of the binding and transport properties of specific proteins in the cell. Finally, we discuss methods to manipulate cellular structural networks by engineering the extracellular environment, including microfabrication of ECM distributions of defined geometry and microdevices designed to measure cellular traction forces at micron-scale resolution. Together, these methods form a powerful arsenal that is already adding significantly to our understanding of the nanoscale architecture and mechanics of living cells and may contribute to the rational design of new cellular micro-and nanotechnologies.     

The Mechanochemical Basis of Cell and Tissue Regulation

This article is a summary of a lecture presented at a symposium on "Mechanics and Chemistry of Biosystems' in honor of Professor Y.C. Fung that convened at the University of California, Irvine in February 2004. The article reviews work from our laboratory that focuses on the mechanism by which mechanical and chemical signals interplay to control how individual cells decide whether to grow, differentiate, move, or die, and thereby promote pattern formation during tissue morphogenesis. Pursuit of this challenge has required development and application of new microtechnologies, theoretical formulations, computational models and bioinformatics tools. These approaches have been used to apply controlled mechanical stresses to specific cell surface molecules and to measure mechanical and biochemical responses; to control cell shape independently of chemical factors; and to handle the structural, hierarchical and informational complexity of living cells. Results of these studies have changed our view of how cells and tissues control their shape and mechanical properties, and have led to the discovery that integrins and the cytoskeleton play a central role in cellular mechanotransduction. Recognition of these critical links between mechanics and cellular biochemistry should lead to novel strategies for the development of new drugs and engineered tissues, as well as biomimetic microdevices and nanotechnologies that more effectively function within the context of living tissues.     

Actuation means for the mechanical stimulation of living cells via microelectromechanical systems: A critical review

Within a living body, cells are constantly exposed to various mechanical constraints. As a matter of fact, these mechanical factors play a vital role in the regulation of the cell state. It is widely recognized that cells can sense, react and adapt themselves to mechanical stimulation. However, investigations aimed at studying cell mechanics directly in vivo remain elusive. An alternative solution is to study cell mechanics via in vitro experiments. Nevertheless, this requires implementing means to mimic the stresses that cells naturally undergo in their physiological environment. In this paper, we survey various microelectromechanical systems (MEMS) dedicated to the mechanical stimulation of living cells. In particular, we focus on their actuation means as well as their inherent capabilities to stimulate a given amount of cells. Thereby, we report actuation means dependent upon the fact they can provide stimulation to a single cell, target a maximum of a hundred cells, or deal with thousands of cells. Intrinsic performances, strengths and limitations are summarized for each type of actuator. We also discuss recent achievements as well as future challenges of cell mechanostimulation.     

Dynamic and structural signatures of lamellar actomyosin force generation

The regulation of cellular traction forces on the extracellular matrix is critical to cell adhesion, migration, proliferation, and differentiation. Diverse lamellar actin organizations ranging from contractile lamellar networks to stress fibers are observed in adherent cells. Although lamellar organization is thought to reflect the extent of cellular force generation, understanding of the physical behaviors of the lamellar actin cytoskeleton is lacking. To elucidate these properties, we visualized the actomyosin dynamics and organization in U2OS cells over a broad range of forces. At low forces, contractile lamellar networks predominate and force generation is strongly correlated to actomyosin retrograde flow dynamics with nominal change in organization. Lamellar networks build ～60% of cellular tension over rapid time scales. At high forces, reorganization of the lamellar network into stress fibers results in moderate changes in cellular tension over slower time scales. As stress fibers build and tension increases, myosin band spacing decreases and α-actinin bands form. On soft matrices, force generation by lamellar networks is unaffected, whereas tension-dependent stress fiber assembly is abrogated. These data elucidate the dynamic and structural signatures of the actomyosin cytoskeleton at different levels of tension and set a foundation for quantitative models of cell and tissue mechanics.     

Trans-scale Granular Modelling of Cytoskeleton: a Mini-Review

Living cells are the functional unit of organs that controls reactions to their exterior. However, the mechanics of living cells can be difficult to characterize due to the crypticity of their microscale structures and associated dynamic cellular processes. Fortunately, multiscale modelling provides a powerful simulation tool that can be used to study the mechanical properties of these soft hierarchical, biological systems. This paper reviews recent developments in hierarchical multiscale modeling technique that aimed at understanding cytoskeleton mechanics. Discussions are expanded with respects to cytoskeletal components including: intermediate filaments, microtubules and microfilament networks. The mechanical performance of difference cytoskeleton components are discussed with respect to their structural and material properties. Explicit granular simulation methods are adopted with different coarse-grained strategies for these cytoskeleton components and the simulation details are introduced in this review.     

Squeezing and detachment of living cells

The interaction of living cells with their environment is linked to their adhesive and elastic properties. Even if the mechanics of simple lipid membranes is fairly well understood, the analysis of single cell experiments remains challenging in part because of the mechanosensory response of cells to their environment. To study the mechanical properties of living cells we have developed a tool that borrows from micropipette aspiration techniques, atomic force microscopy, and the classical Johnson-Kendall-Roberts test. We show results from a study of the adhesion properties of living cells, as well as the elastic response and relaxation. We present models that are applied throughout the different stages of an experiment, which indicate that the contribution of the different components of the cell are active at various stages of compression, retraction, and detachment. Finally, we present a model that attempts to elucidate the surprising logarithmic relaxation observed when the cell is subjected to a given deformation.     

Probing Cell Structure Responses Through a Shear and Stretching Mechanical Stimulation Technique

Cells are complex, dynamic systems that respond to various in vivo stimuli including chemical, mechanical, and scaffolding alterations. The influence of mechanics on cells is especially important in physiological areas that dictate what modes of mechanics exist. Complex, multivariate physiological responses can result from multi-factorial, multi-mode mechanics, including tension, compression, or shear stresses. In this study, we present a novel device based on elastomeric materials that allowed us to stimulate NIH 3T3 fibroblasts through uniaxial strip stretching or shear fluid flow. Cell shape and structural response was observed using conventional approaches such as fluorescent microscopy. Cell orientation and actin cytoskeleton alignment along the direction of applied force were observed to occur after an initial 3 h time period for shear fluid flow and static uniaxial strip stretching experiments although these two directions of alignment were oriented orthogonal relative to each other. This response was then followed by an increasingly pronounced cell and actin cytoskeleton alignment parallel to the direction of force after 6, 12, and 24 h, with 85% of the cells aligned along the direction of force after 24 h. These results indicate that our novel device could be implemented to study the effects of multiple modes of mechanical stimulation on living cells while probing their structural response especially with respect to competing directions of alignment and orientation under these different modes of mechanical stimulation. We believe that this will be important in a diversity of fields including cell mechanotransduction, cell–material interactions, biophysics, and tissue engineering.     

Relationship between cell stiffness and stress fiber amount,assessed by simultaneous atomic force microscopy and live-cell fluorescence imaging

Actomyosin stress fibers, one of the main components of the cell’s cytoskeleton, provide mechanical stability to adherent cells by applying and transmitting tensile forces onto the extracellular matrix (ECM) at the sites of cell–ECM adhesion. While it is widely accepted that changes in spatial and temporal distribution of stress fibers affect the cell’s mechanical properties, there is no quantitative knowledge on how stress fiber amount and organization directly modulate cell stiffness. We address this key open question by combining atomic force microscopy with simultaneous fluorescence imaging of living cells, and combine for the first time reliable quantitative parameters obtained from both techniques. We show that the amount of myosin and (to a lesser extent) actin assembled in stress fibers directly modulates cell stiffness in adherent mouse fibroblasts (NIH3T3). In addition, the spatial distribution of stress fibers has a second-order modulatory effect. In particular, the presence of either fibers located in the cell periphery, aligned fibers or thicker fibers gives rise to reinforced cell stiffness. Our results provide basic and significant information that will help design optimal protocols to regulate the mechanical properties of adherent cells via pharmacological interventions that alter stress fiber assembly or via micropatterning techniques that restrict stress fiber spatial organization.     

Time-Dependent Micromechanical Responses of Breast Cancer Cells and Adjacent Fibroblasts to Electric Treatment

Direct-current, low-intensity, electric fields were suggested as a minimally invasive treatment for various cancers. The tumor microenvironment may affect treatment efficacy, albeit it has not generally been considered when evaluating novel anti-cancer treatments. We evaluate the effects of electric treatment on epithelial, breast-cancer cells, co-cultured with non-cancerous fibroblasts, a simplified model for the tumor-microenvironment. We evaluate changes in morphology, cytoskeleton, and focus on dynamic intracellular structure and mechanics. Multiple-particle tracking was used within living cells to quantify time-dependent structural and mechanical changes. Cancer cells suffer severe cell death and exhibit transient rounding and changes in internal structural and mechanics. Interestingly, treating cancer cells in co-culture with fibroblasts delays and reduces their responses to treatment. Our particle-tracking data indicates a mechanism relating the observed changes in intracellular transport to transient changes in the microtubule network and its motors. In contrast, fibroblasts are only minimally affected by treatment, separately or in co-culture. To conclude, intracellular mechanics reveal time-dependent responses after treatment, unavailable by bulk measurements. This time-dependence could provide a window of opportunity for continued treatment. We demonstrate the importance of evaluating anti-cancer treatments within their microenvironment, which can affect response magnitude and time-course.     

The role of prestress and architecture of the cytoskeleton and deformability of cytoskeletal filaments in mechanics of adherent cells: a quantitative analysis.

Mechanical properties of adherent cells were investigated using methods of engineering mechanics. The cytoskeleton (CSK) was modeled as a filamentous network and key mechanisms and corresponding molecular structures which determine cell elastic behavior were identified. Three models of the CSK were considered: open-cell foam networks, prestressed cable nets, and a tensegrity model as a special case of the latter. For each model, the modulus of elasticity (i.e. an index of resistance to small deformation) was given as a function of mechanical and geometrical properties of CSK filaments whose values were determined from the data in the literature. Quantitative predictions for the elastic modulus were compared with data obtained previously from mechanical tests on adherent cells. The open-cell foam model yielded the elastic modulus (10(3)-10(4)Pa) which was consistent with measurements which apply a large compressive stress to the cell. This suggests that bending of CSK filaments is the key mechanism for resisting large compression. The prestressed cable net and tensegrity model yielded much lower elastic moduli (10(1)-10(2)Pa) which were consistent with values determined from equilibrium measurements at low applied stress. This suggests that CSK prestress and architecture are the primary determinants of the cell elastic response. The tensegrity model revealed the possibility that buckling of microtubules of the CSK also contributed to cell elasticity.     

Mathematical modelling of the cellular mechanics of plants

The complex mechanical behaviour of plant tissues reflects the complexity of their structure and material properties. Modelling has been widely used in studies of how cell walls, single cells and tissue respond to loading, both externally applied loading and loads on the cell wall resulting from changes in the pressure within fluid-filled cells. This paper reviews what approaches have been taken to modelling and simulation of cell wall, cell and tissue mechanics, and to what extent models have been successful in predicting mechanical behaviour. Advances in understanding of cell wall ultrastructure and the control of cell growth present opportunities for modelling to clarify how growth-related mechanical properties arise from wall polymeric structure and biochemistry.     

In the middle of it all: Mutual mechanical regulation between the nucleus and the cytoskeleton

The nucleus is typically treated as the large phase-dense or easy-to-label structure at the center of the cell which is manipulated by the governing mechanical machinery inside the cytoplasm. However, recent evidence has suggested that the mechanical properties of the nucleus are important to cell fate. We will discuss many aspects of the structural and functional interconnections between nuclear mechanics and cellular mechanics in this review. There are numerous implications for the progression of many disease states associated with both nuclear structural proteins and cancers. The nucleus itself is a large organelle taking up significant volume within the cell, and most studies agree that nuclei are significantly stiffer than the surrounding cytoplasm. Thus when a cell is exposed to force, the nucleus is exposed to and helps resist that force. The nucleus and nucleoskeleton are interconnected with the cellular cytoskeleton, and these connections may aid in helping disperse forces within tissues and/or with mechanotransduction. During translocation and transmigration the nucleus can act as a resistive element. Understanding the role of mechanical regulation of the nucleus may aid in understanding cellular motility and crawling through confined geometries. Thus the nucleus plays a role in developing mechanical territories and niches, affecting rates of wound healing and allowing cells to transmigrate through tissues for developmental, repair or pathological means.