A prestressed cable network model of the adherent cell cytoskeleton

Subcellular mechano-regulation of cell migration in confined extracellular microenvironment

Cell migration is a highly coordinated cellular event that determines diverse physiological and pathological processes in which the continuous interaction of a migrating cell with neighboring cells or the extracellular matrix is regulated by the physical setting of the extracellular microenvironment. In confined spaces, cell migration occurs differently compared to unconfined open spaces owing to the additional forces that limit cell motility, which create a driving bias for cells to invade the confined space, resulting in a distinct cell motility process compared to what is expected in open spaces. Moreover, cells in confined environments can be subjected to elevated mechanical compression, which causes physical stimuli and activates the damage repair cycle in the cell, including the DNA in the nucleus. Although cells have a self-restoring system to repair damage from the cell membrane to the genetic components of the nucleus, this process may result in genetic and/or epigenetic alterations that can increase the risk of the progression of diverse diseases, such as cancer and immune disorders. Furthermore, there has been a shift in the paradigm of bioengineering from the development of new biomaterials to controlling biophysical cues and fine-tuning cell behaviors to cure damaged/diseased tissues. The external physical cues perceived by cells are transduced along the mechanosensitive machinery, which is further channeled into the nucleus through subcellular molecular linkages of the nucleoskeleton and cytoskeleton or the biochemical translocation of transcription factors. Thus, external cues can directly or indirectly regulate genetic transcriptional processes and nuclear mechanics, ultimately determining cell fate. In this review, we discuss the importance of the biophysical cues, response mechanisms, and mechanical models of cell migration in confined environments. We also discuss the effect of force-dependent deformation of subcellular components, specifically focusing on subnuclear organelles, such as nuclear membranes and chromosomal organization. This review will provide a biophysical perspective on cancer progression and metastasis as well as abnormal cellular proliferation.

Study of the Effect of Cell Prestress on the Cell Membrane Penetration Behavior by Atomic Force Microscopy

In recent years, atomic force microscopes have been used for cell transfection because of their high-precision micro-indentation mode; however, the insertion efficiency of the tip of AFM into cells is extremely low. In this study, NIH3T3 mouse fibroblast cells cultured on a flexible dish with micro-groove patterns were subjected to various substrate strains at 5%, 10%, 15%, and 20%. It was found that the cell stiffness depends on the prestress of the cell membrane, and that the insertion rate of AFM tips into the cell membrane is proportional to the stiffness through the AFM indentation experiment. The finite element analysis proves that prestress increases the bending stiffness of the cytoskeleton, allowing it to better support the cell membrane, which realizes the stress concentration in the contact area between the AFM tip and the cell membrane. The results indicate that the prestress contributes to the mechanical properties of the cell and suggest that the insertion efficiency could be greatly improved with an increase of the prestress of the cell membrane.

Valve interstitial cells under impact load, a mechanobiology study

Understanding the relationship between mechanobiology and the biosynthetic activities of the valve interstitial cells (VICs) in health and disease under severe dynamic loading conditions is of particular interest. The purpose of this study is to further understand the mechanobiology of heart valve leaflet tissue and the VICs under impact forces. Two novel computational and experimental platforms were developed to study the effect of impact load on the VICs to monitor for apoptosis. The first objective was to design and develop an apparatus to experimentally study viability (apoptosis) of the porcine heart valve leaflet tissue VICs in the aortic position under controlled impact forces. Apoptosis was assessed based on terminal transferase dUTP nick end-labelling (TUNEL) assay. The second objective was to develop a computational platform to estimate the stress and strain fields in the vicinity of VICs when the tissue experiences impact forces. A nonlinear finite element (FE) model with an anisotropic, hyperelastic and heterogeneous material model for the matrix and cells was developed. Preliminary results confirm that interstitial cells are successfully resistant to impact loads up to 30 times more than normal physiological conditions. Additionally, the structure and composition of heart valve leaflet tissue provides a mechanical shield for VICs protecting them from excessive mechanical forces such as impact loads. Although, the entire tissue may experience excessive stresses, which may lead to structural damage, the stresses around and near VICs remain consistency low. Results of this study may be used for heart valve leaflet tissue-engineering, as well as further understanding the mechanobiology of the VICs in health and disease.

Biomechanical characterization and modeling of human mesenchymal stem cells under compression

The application of microelectromechanical systems (MEMS) in biomedical devices has expanded vastly over the last few decades, with MEMS devices being developed to measure different characteristics of cells. The study of cell mechanics offers valuable understanding of cell viability and functionality. Cell biomechanics approaches also facilitate the characterization of important cell and tissue behaviors. In particular, understanding of the biological response of cells to their biomechanical environment would enhance the knowledge of how cellular responses correlate to tissue level characteristics and how some diseases, such as cancer, grow in the body. This study focuses on viscoelastic modeling of the behavior of a single suspended human mesenchymal stem cell (hMSC). Mechanical properties of hMSC cells are particularly important in tissue engineering and research for the treatment of cardiovascular diseases. We evaluated the elastic and viscoelastic properties of hMSC cells using a miniaturized custom-made BioMEMS device. Our results were compared to the elastic and viscoelastic properties measured by other methods such as atomic force microscopy (AFM) and micropipette aspiration. Different approaches were applied to model the experimentally obtained force data, including elastic and Standard Linear Solid (SLS) constitutive models, and the corresponding constants were derived. These values were compared to the ones in literature that were based on micropipette aspiration and AFM methods. We then utilized a tensegrity approach to model major parts of the internal structure of the cell and treat the cell as a network of viscoelastic microtubules and microfilaments, as opposed to a simple spherical blob. The results predicted from the tensegrity model were similar to the recorded experimental data.

Biomechanics of cells and subcellular components: A comprehensive review of computational models and applications

Cells are a fundamental structural, functional and biological unit for all living organisms. Up till now, considerable efforts have been made to study the responses of single cells and subcellular components to an external load, and understand the biophysics underlying cell rheology, mechanotransduction and cell functions using experimental and in silico approaches. In the last decade, computational simulation has become increasingly attractive due to its critical role in interpreting experimental data, analysing complex cellular/subcellular structures, facilitating diagnostic designs and therapeutic techniques, and developing biomimetic materials. Despite the significant progress, developing comprehensive and accurate models of living cells remains a grand challenge in the 21st century. To understand current state of the art, this review summarises and classifies the vast array of computational biomechanical models for cells. The article covers the cellular components at multi-spatial levels, that is, protein polymers, subcellular components, whole cells and the systems with scale beyond a cell. In addition to the comprehensive review of the topic, this article also provides new insights into the future prospects of developing integrated, active and high-fidelity cell models that are multiscale, multi-physics and multi-disciplinary in nature. This review will be beneficial for the researchers in modelling the biomechanics of subcellular components, cells and multiple cell systems and understanding the cell functions and biological processes from the perspective of cell mechanics.

More than just a barrier: using physical models to couple membrane shape to cell function

The correct execution of many cellular processes, such as division and motility, requires the cell to adopt a specific shape. Physically, these shapes are determined by the interplay of the plasma membrane and internal cellular driving factors. While the plasma membrane defines the boundary of the cell, processes inside the cell can result in the generation of forces that deform the membrane. These processes include protein binding, the assembly of protein superstructures, and the growth and contraction of cytoskeletal networks. Due to the complexity of the cell, relating observed membrane deformations back to internal processes is a challenging problem. Here, we review cell shape changes in endocytosis, cell adhesion, cell migration and cell division and discuss how by modeling membrane deformations we can investigate the inner working principles of the cell.

Altered mechanical properties of actin fibers due to breast cancer invasion: parameter identification based on micropipette aspiration and multiscale tensegrity modeling

The biophysical properties of cells change with cancer invasion to fulfill their metastatic behavior. Cell softening induced by cancer is highly associated with alterations in cytoskeleton fibers. Changes in the mechanical properties of cytoskeletal fibers have not been quantified due to technical limitations. In this study, we used the micropipette aspiration technique to calculate and compare the viscoelastic properties of non-invasive and invasive breast cancer cells. We evaluated the mechanical properties of actin fibers and microtubules of two cancerous cell lines by using multiscale tensegrity modeling and an optimization method. Cancer invasion caused altered viscoelastic behavior of cells and the results of modeling showed changes in mechanical properties of major cytoskeleton fibers. The stiffness and viscosity constant of actin fibers in non-invasive cells were 1.28 and 2.27 times higher than those of the invasive cells, respectively. However, changes in mechanical properties of microtubules were minor. Immunofluorescent staining of fibers and their quantified distributions confirmed altered actin distribution among two cell lines, in contrast to microtubule distribution. This study highlights the function of cytoskeletal fibers in cancer progression, which could be of interest in designing therapeutic strategies to target cancer progress. Firstly, the viscoelastic behavior of non-invasive and invasive cells is examined with micropipette aspiration tests. A tensegrity model of cells is developed to mimic the viscoelastic behavior of cells, and tensegrity element stiffness is evaluated in an optimization procedure based on micropipette aspiration tests. Finally, by using immunofluorescent staining and confocal imaging, mechanical properties of actin filaments and microtubules of cancer cells are investigated during the course of metastasis.

Migration of the 3T3 Cell with a Lamellipodium on Various Stiffness Substrates—Tensegrity Model

Changes in mechanical stimuli and the physiological environment are sensed by the cell. Thesechanges influence the cell’s motility patterns. The cell’s directional migration is dependent on the substrate stiffness. To describe such behavior of a cell, a tensegrity model was used. Cells with an extended lamellipodium were modeled. The internal elastic strain energy of a cell attached to the substrates with different stiffnesses was evaluated. The obtained results show that on the stiffer substrate, the elastic strain energy of the cell adherent to this substrate decreases. Therefore, the substrate stiffness is one of the parameters that govern the cell’s directional movement.

Nanoscopic Approach to Study the Early Stages of Epithelial to Mesenchymal Transition (EMT) of Human Retinal Pigment Epithelial (RPE) Cells In Vitro

The maintenance of visual function is supported by the proper functioning of the retinal pigment epithelium (RPE), representing a mosaic of polarized cuboidal postmitotic cells. Damage factors such as inflammation, aging, or injury can initiate the migration and proliferation of RPE cells, whereas they undergo a pseudo-metastatic transformation or an epithelial to mesenchymal transition (EMT) from cuboidal epithelioid into fibroblast-like or macrophage-like cells. This process is recognized as a key feature in several severe ocular pathologies, and is mimicked by placing RPE cells in culture, which provides a reasonable and well-characterized in vitro model for a type 2 EMT. The most obvious characteristic of EMT is the cell phenotype switching, accompanied by the cytoskeletal reorganization with changes in size, shape, and geometry. Atomic force microscopy (AFM) has the salient ability to label-free explore these characteristics. Based on our AFM results supported by the genetic analysis of specific RPE differentiation markers, we elucidate a scheme for gradual transformation from the cobblestone to fibroblast-like phenotype. Structural changes in the actin cytoskeletal reorganization at the early stages of EMT lead to the development of characteristic geodomes, a finding that may reflect an increased propensity of RPE cells to undergo further EMT and thus become of diagnostic significance.

An Active Biomechanical Model of Cell Adhesion Actuated by Intracellular Tensioning-Taxis

Cell adhesion to the extracellular matrix (ECM) is highly active and plays a crucial role in various physiological functions. The active response of cells to physicochemical cues has been universally discovered in multiple microenvironments. However, the mechanisms to rule these active behaviors of cells are still poorly understood. Here, we establish an active model to probe the biomechanical mechanisms governing cell adhesion. The framework of cells is modeled as a tensional integrity that is maintained by cytoskeletons and extracellular matrices. Active movement of the cell model is self-driven by its intrinsic tendency to intracellular tensioning, defined as tensioning-taxis in this study. Tensioning-taxis is quantified as driving potential to actuate cell adhesion, and the traction forces are solved by our proposed numerical method of local free energy adaptation. The modeling results account for the active adhesion of cells with dynamic protruding of leading edge and power-law development of mechanical properties. Furthermore, the morphogenesis of cells evolves actively depending on actin filaments alignments by a predicted mechanism of scaling and directing traction forces. The proposed model provides a quantitative way to investigate the active mechanisms of cell adhesion and holds the potential to guide studies of more complex adhesion and motion of cells coupled with multiple external cues.

Modeling cell migration regulated by cell extracellular-matrix micromechanical coupling

Cell migration in fibrous extracellular matrix (ECM) is crucial to many physiological and pathological processes such as tissue regeneration, immune response, and cancer progression. During migration, individual cells can generate active pulling forces via actomyosin contraction, which are transmitted to the ECM fibers through focal adhesion complexes, remodel the ECM, and eventually propagate to and can be sensed by other cells in the system. The microstructure and physical properties of the ECM can also significantly influence cell migration, e.g., via durotaxis and contact guidance. Here, we develop a computational model for two-dimensional cell migration regulated by cell-ECM micromechanical coupling. Our model explicitly takes into account a variety of cellular-level processes, including focal adhesion formation and disassembly, active traction force generation and cell locomotion due to actin filament contraction, transmission and propagation of tensile forces in the ECM, as well as the resulting ECM remodeling. We validate our model by accurately reproducing single-cell dynamics of MCF-10A breast cancer cells migrating on collagen gels and show that the durotaxis and contact guidance effects naturally arise as a consequence of the cell-ECM micromechanical interactions considered in the model. Moreover, our model predicts strongly correlated multicellular migration dynamics, which are resulted from the ECM-mediated mechanical coupling among the migrating cell and are subsequently verified in in vitro experiments using MCF-10A cells. Our computational model provides a robust tool to investigate emergent collective dynamics of multicellular systems in complex in vivo microenvironment and can be utilized to design in vitro microenvironments to guide collective behaviors and self-organization of cells.

Substrate stiffness and mechanical stress due to intercellular cooperativity guides tissue structure

A key challenge in cell and tissue morphogenesis is to understand how a crucial balance between cell proliferation and apoptosis maintains an evolving tissue structure. These processes are mutually non-exclusive and require stiffness monitoring of the host substrate. Adhered cells actively mechanosense the tension in the extracellular matrix (ECM). They collectively alter self-organization and generate a host of tissue patterns. Using an in silico elastic fiber-network in two dimensions, we simulate cell-ECM composite structures and characterize features of the emerging tissue patterns during successive cell proliferation and apoptosis. Our data reveals that, in general, cell viability is a function of the cell-induced effective ECM stiffness supported by intercellular cooperativity. Translating this into a remodeling tissue, we find that average cell cycle duration in concert with the locally stressed regions of the ECM promote heterogeneous proliferation and apoptosis inducing finger-like protrusions along the tissue periphery - a feature normally observed during tumorigenesis. Further, we find that recovery of a scratch wound is delayed for cells harbored on a compliant or (and) in a highly collagen depleted ECM.

Automatic directional analysis of cell fluorescence images and morphological modeling of microfilaments

Cytoskeleton and nucleus are two important anatomic components in eukaryotic cells. Cell fluorescence images are employed to study their realignment and deformation during cell extrusion. Quantitative analysis and modeling of cell orientation are investigated in this paper. For orientation measurement, alignment orientation of microfilaments is calculated using structure tensor method. Nuclei is segmented and fitted to ellipses in nuclei images. Based on the fitted ellipse, orientation and aspect ratio of each nucleus are computed. A morphological model is proposed to describe the movement of microfilaments quantitatively. The parameters of the model are determined by in-plane stresses obtained by numerical simulation. The proposed automatic orientation measurement algorithms can help to analyze the relationship between cell orientation and stress qualitatively. The proposed morphological model is the first model to quantitatively describe the relationship of microfilament movement with stress. Experimental results show that cell and nucleus tend to align along in-plane maximum shear stress and the proposed morphological model is a reasonable model for cell movement. The modeling of cell behavior under different stress can facilitate biomedical research such as tissue engineering and cancer analysis. Graphical abstract ᅟ.

Empirically Determined Vascular Smooth Muscle Cell Mechano-Adaptation Law

Cardiovascular disease can alter the mechanical environment of the vascular system, leading to mechano-adaptive growth and remodeling. Predictive models of arterial mechano-adaptation could improve patient treatments and outcomes in cardiovascular disease. Vessel-scale mechano-adaptation includes remodeling of both the cells and extracellular matrix. Here, we aimed to experimentally measure and characterize a phenomenological mechano-adaptation law for vascular smooth muscle cells (VSMCs) within an artery. To do this, we developed a highly controlled and reproducible system for applying a chronic step-change in strain to individual VSMCs with in vivo like architecture and tracked the temporal cellular stress evolution. We found that a simple linear growth law was able to capture the dynamic stress evolution of VSMCs in response to this mechanical perturbation. These results provide an initial framework for development of clinically relevant models of vascular remodeling that include VSMC adaptation.

Modeling Living Cells Response to Surface Tension and Chemical Patterns

Mechanobiology is an important epigenetic factor. It influences cell functioning and bears on gene induction, protein synthesis, cell growth, and differentiation. In the presence of patterned chemical cues, living cells can take shapes that are far from that of a drop of fluid. These shapes are characterized by inward curvatures that are pinned at the points of location of the cues. The mechanochemical interactions that orchestrate cell behavior is simulated and controlled by modeling the cells as made by parcels of fluid. Cells become drops that are then endowed with the presence of additional forces, generated on the fly, that effectively make them active. With the proper choice of the forces, the phenomena that emerge from the dynamics match quantitatively the experiments. A combination of hydrophilic and lipophilic forces acting between the beads of fluid allows the active drop to respond to patterned cues and form squares, pentagons, hexagons, and flowers, just as living cells do.

Modeling cell shape and dynamics on micropatterns

Adhesive micropatterns have become a standard tool to study cells under defined conditions. Applications range from controlling the differentiation and fate of single cells to guiding the collective migration of cell sheets. In long-term experiments, single cell normalization is challenged by cell division. For all of these setups, mathematical models predicting cell shape and dynamics can guide pattern design. Here we review recent advances in predicting and explaining cell shape, traction forces and dynamics on micropatterns. Starting with contour models as the simplest approach to explain concave cell shapes, we move on to network and continuum descriptions as examples for static models. To describe dynamic processes, cellular Potts, vertex and phase field models can be used. Different types of model are appropriate to address different biological questions and together, they provide a versatile tool box to predict cell behavior on micropatterns.

Modeling the mechanics of cells in the cell-spreading process driven by traction forces

Mechanical properties of cells and their mechanical interaction with the extracellular environments are main factors influencing cellular function, thus indicating the progression of cells in different disease states. By considering the mechanical interactions between cell adhesion molecules and the extracellular environment, we developed a cell mechanical model that can characterize the mechanical changes in cells during cell spreading. A cell model was established that consisted of various main subcellular components, including cortical cytoskeleton, nuclear envelope, actin filaments, intermediate filaments, and microtubules. We demonstrated the structural changes in subcellular components and the changes in spreading areas during cell spreading driven by traction forces. The simulation of nanoindentation tests was conducted by integrating the indenting force to the cell model. The force-indentation curve of the cells at different spreading states was simulated, and the results showed that cell stiffness increased with increasing traction forces, which were consistent with the experimental results. The proposed cell mechanical model provides a strategy to investigate the mechanical interactions of cells with the extracellular environments through the adhesion molecules and to reveal the cell mechanical properties at the subcellular level as cells shift from the suspended state to the adherent state.

Deformability-based circulating tumor cell separation with conical-shaped microfilters: Concept, optimization, and design criteria

Circulating tumor cells (CTCs) separation technology has made positive impacts on cancer science in many aspects. The ability of detecting and separating CTCs can play a key role in early cancer detection and treatment. In recent years, there has been growing interest in using deformability-based CTC separation microfilters due to their simplicity and low cost. Most of the previous studies in this area are mainly based on experimental work. Although experimental research provides useful insights in designing CTC separation devices, there is still a lack of design guidelines based on fundamental understandings of the cell separation process in the filters. While experimental efforts face challenges, especially microfabrication difficulties, we adopt numerical simulation here to study conical-shaped microfilters using deformability difference between CTCs and blood cells for the separation process. We use the liquid drop model for modeling a CTC passing through such microfilters. The accuracy of the model in predicting the pressure signature of the system is validated by comparing it with previous experiments. Pressure-deformability analysis of the cell going through the channel is then carried out in detail in order to better understand how a CTC behaves throughout the filtration process. Different system design criteria such as system throughput and unclogging of the system are discussed. Specifically, pressure behavior under different system throughput is analyzed. Regarding the unclogging issue, we define pressure ratio as a key parameter representing the ability to overcome clogging in such CTC separation devices and investigate the effect of conical angle on the optimum pressure ratio. Finally, the effect of unclogging applied pressure on the system performance is examined. Our study provides detailed understandings of the cell separation process and its characteristics, which can be used for developing more efficient CTC separation devices.

A quantitative quantum-chemical analysis tool for the distribution of mechanical force in molecules

The promising field of mechanochemistry suffers from a general lack of understanding of the distribution and propagation of force in a stretched molecule, which limits its applicability up to the present day. In this article, we introduce the JEDI (Judgement of Energy DIstribution) analysis, which is the first quantum chemical method that provides a quantitative understanding of the distribution of mechanical stress energy among all degrees of freedom in a molecule. The method is carried out on the basis of static or dynamic calculations under the influence of an external force and makes use of a Hessian matrix in redundant internal coordinates (bond lengths, bond angles, and dihedral angles), so that all relevant degrees of freedom of a molecule are included and mechanochemical processes can be interpreted in a chemically intuitive way. The JEDI method is characterized by its modest computational effort, with the calculation of the Hessian being the rate-determining step, and delivers, except for the harmonic approximation, exact ab initio results. We apply the JEDI analysis to several example molecules in both static quantum chemical calculations and Born-Oppenheimer Molecular Dynamics simulations in which molecules are subject to an external force, thus studying not only the distribution and the propagation of strain in mechanically deformed systems, but also gaining valuable insights into the mechanochemically induced isomerization of trans-3,4-dimethylcyclobutene to trans,trans-2,4-hexadiene. The JEDI analysis can potentially be used in the discussion of sonochemical reactions, molecular motors, mechanophores, and photoswitches as well as in the development of molecular force probes.

Kinetics of virus entry by endocytosis

Entry of virions into the host cells is either endocytotic or fusogenic. In both cases, it occurs via reversible formation of numerous relatively weak bonds resulting in wrapping of a virion by the host membrane with subsequent membrane rupture or scission. The corresponding kinetic models are customarily focused on the formation of bonds and do not pay attention to the energetics of the whole process, which is crucially dependent, especially in the case of endocytosis, on deformation of actin filaments forming the cytoskeleton of the host cell. The kinetic model of endocytosis, proposed by the author, takes this factor into account and shows that the whole process can be divided into a rapid initial transient stage and a long steady-state stage. The entry occurs during the latter stage and can be described as a first-order reaction. Depending on the details of the dependence of the grand canonical potential on the number of bonds, the entry can be limited either by the interplay of bond formation and membrane rupture (or scission) or by reaching a maximum of this potential.

Dynamics of cell shape and forces on micropatterned substrates predicted by a cellular Potts model

Micropatterned substrates are often used to standardize cell experiments and to quantitatively study the relation between cell shape and function. Moreover, they are increasingly used in combination with traction force microscopy on soft elastic substrates. To predict the dynamics and steady states of cell shape and forces without any a priori knowledge of how the cell will spread on a given micropattern, here we extend earlier formulations of the two-dimensional cellular Potts model. The third dimension is treated as an area reservoir for spreading. To account for local contour reinforcement by peripheral bundles, we augment the cellular Potts model by elements of the tension-elasticity model. We first parameterize our model and show that it accounts for momentum conservation. We then demonstrate that it is in good agreement with experimental data for shape, spreading dynamics, and traction force patterns of cells on micropatterned substrates. We finally predict shapes and forces for micropatterns that have not yet been experimentally studied.

The spatial-temporal characteristics of type I collagen-based extracellular matrix

Type I collagen abounds in mammalian extracellular matrix (ECM) and is crucial to many biophysical processes. While previous studies have mostly focused on bulk averaged properties, here we provide a comprehensive and quantitative spatial-temporal characterization of the microstructure of type I collagen-based ECM as the gelation temperature varies. The structural characteristics including the density and nematic correlation functions are obtained by analyzing confocal images of collagen gels prepared at a wide range of gelation temperatures (from 16 °C to 36 °C). As temperature increases, the gel microstructure varies from a "bundled" network with strong orientational correlation between the fibers to an isotropic homogeneous network with no significant orientational correlation, as manifested by the decaying of length scales in the correlation functions. We develop a kinetic Monte-Carlo collagen growth model to better understand how ECM microstructure depends on various environmental or kinetic factors. We show that the nucleation rate, growth rate, and an effective hydrodynamic alignment of collagen fibers fully determines the spatiotemporal fluctuations of the density and orientational order of collagen gel microstructure. Also the temperature dependence of the growth rate and nucleation rate follow the prediction of classical nucleation theory.

In situ determination of alveolar septal strain, stress and effective Young's modulus: an experimental/computational approach

Alveolar septa, which have often been modeled as linear elements, may distend due to inflation-induced reduction in slack or increase in tissue stretch. The distended septum supports tissue elastic and interfacial forces. An effective Young's modulus, describing the inflation-induced relative displacement of septal end points, has not been determined in situ for lack of a means of determining the forces supported by septa in situ. Here we determine such forces indirectly according to Mead, Takishima, and Leith's classic lung mechanics analysis (J Appl Physiol 28: 596-608, 1970), which demonstrates that septal connections transmit the transpulmonary pressure, PTP, from the pleural surface to interior regions. We combine experimental septal strain determination and computational stress determination, according to Mead et al., to calculate effective Young's modulus. In the isolated, perfused rat lung, we label the perfusate with fluorescence to visualize the alveolar septa. At eight PTP values around a ventilation loop between 4 and 25 cmH2O, and upon total deflation, we image the same region by confocal microscopy. Within an analysis region, we measure septal lengths. Normalizing by unstressed lengths at total deflation, we calculate septal strains for all PTP > 0 cmH2O. For the static imaging conditions, we computationally model application of PTP to the boundary of the analysis region and solve for septal stresses by least squares fit of an overdetermined system. From group septal strain and stress values, we find effective septal Young's modulus to range from 1.2 × 10(5) dyn/cm(2) at low P(TP) to 1.4 × 10(6) dyn/cm(2) at high P(TP).

Tensegrity, cellular biophysics, and the mechanics of living systems

The recent convergence between physics and biology has led many physicists to enter the fields of cell and developmental biology. One of the most exciting areas of interest has been the emerging field of mechanobiology that centers on how cells control their mechanical properties, and how physical forces regulate cellular biochemical responses, a process that is known as mechanotransduction. In this article, we review the central role that tensegrity (tensional integrity) architecture, which depends on tensile prestress for its mechanical stability, plays in biology. We describe how tensional prestress is a critical governor of cell mechanics and function, and how use of tensegrity by cells contributes to mechanotransduction. Theoretical tensegrity models are also described that predict both quantitative and qualitative behaviors of living cells, and these theoretical descriptions are placed in context of other physical models of the cell. In addition, we describe how tensegrity is used at multiple size scales in the hierarchy of life—from individual molecules to whole living organisms—to both stabilize three-dimensional form and to channel forces from the macroscale to the nanoscale, thereby facilitating mechanochemical conversion at the molecular level.

Physical aspects of the initial phase of endocytosis

The endocytotic mechanism of entry of virions into cells includes wrapping of a virion by the host membrane with subsequent formation of a vesicle covering a virion. The energy along this pathway depends on the ligand-receptor interaction and deformation of the cell membrane and underlying actin cytoskeleton. The available models describe the cytoskeleton deformation by using the conventional continuum theory of elasticity and predict that this factor often controls the repulsive part of the virion-cell interaction. This approach is, however, debatable because the size of virions is smaller than or comparable to the length scale characterizing the cytoskeleton structure. The author shows that the continuum theory appreciably (up to one order of magnitude) overestimates the cytoskeleton-deformation energy and that the scale of this energy is comparable to that of cell membrane bending.

Form-finding model shows how cytoskeleton network stiffness is realized

In eukaryotic cells the actin-cytoskeletal network provides stiffness and the driving force that contributes to changes in cell shape and cell motility, but the elastic behavior of this network is not well understood. In this paper a two dimensional form-finding model is proposed to investigate the elasticity of the actin filament network. Utilizing an initially random array of actin filaments and actin-cross-linking proteins the form-finding model iterates until the random array is brought into a stable equilibrium configuration. With some care given to actin filament density and length, distance between host sites for cross-linkers, and overall domain size the resulting configurations from the form-finding model are found to be topologically similar to cytoskeletal networks in real cells. The resulting network may then be mechanically exercised to explore how the actin filaments deform and align under load and the sensitivity of the network’s stiffness to actin filament density, length, etc. Results of the model are consistent with the experimental literature, e.g. actin filaments tend to re-orient in the direction of stretching; and the filament relative density, filament length, and actin-cross-linking protein’s relative density, control the actin-network stiffness. The model provides a ready means of extension to more complicated domains and a three-dimensional form-finding model is under development as well as models studying the formation of actin bundles.

Computational aspects in mechanical modeling of the articular cartilage tissue

This review focuses on the modeling of articular cartilage (at the tissue level), chondrocyte mechanobiology (at the cell level) and a combination of both in a multiscale computation scheme. The primary objective is to evaluate the advantages and disadvantages of conventional models implemented to study the mechanics of the articular cartilage tissue and chondrocytes. From monophasic material models as the simplest form to more complicated multiscale theories, these approaches have been frequently used to model articular cartilage and have contributed significantly to modeling joint mechanics, addressing and resolving numerous issues regarding cartilage mechanics and function. It should be noted that attentiveness is important when using different modeling approaches, as the choice of the model limits the applications available. In this review, we discuss the conventional models applicable to some of the mechanical aspects of articular cartilage such as lubrication, swelling pressure and chondrocyte mechanics and address some of the issues associated with the current modeling approaches. We then suggest future pathways for a more realistic modeling strategy as applied for the simulation of the mechanics of the cartilage tissue using multiscale and parallelized finite element method.

Contractile network models for adherent cells

Cells sense the geometry and stiffness of their adhesive environment by active contractility. For strong adhesion to flat substrates, two-dimensional contractile network models can be used to understand how force is distributed throughout the cell. Here we compare the shape and force distribution for different variants of such network models. In contrast to Hookean networks, cable networks reflect the asymmetric response of biopolymers to tension versus compression. For passive networks, contractility is modeled by a reduced resting length of the mechanical links. In actively contracting networks, a constant force couple is introduced into each link in order to model contraction by molecular motors. If combined with fixed adhesion sites, all network models lead to invaginated cell shapes, but only actively contracting cable networks lead to the circular arc morphology typical for strongly adhering cells. In this case, shape and force distribution are determined by local rather than global determinants and thus are suited to endow the cell with a robust sense of its environment. We also discuss nonlinear and adaptive linker mechanics as well as the relation to tissue shape.

Viscoelastic response of contractile filament bundles

The actin cytoskeleton of adherent tissue cells often condenses into filament bundles contracted by myosin motors, so-called stress fibers, which play a crucial role in the mechanical interaction of cells with their environment. Stress fibers are usually attached to their environment at the endpoints, but possibly also along their whole length. We introduce a theoretical model for such contractile filament bundles which combines passive viscoelasticity with active contractility. The model equations are solved analytically for two different types of boundary conditions. A free boundary corresponds to stress fiber contraction dynamics after laser surgery and results in good agreement with experimental data. Imposing cyclic varying boundary forces allows us to calculate the complex modulus of a single stress fiber.

Stress transmission within the cell

An outstanding problem in cell biology is how cells sense mechanical forces and how those forces affect cellular functions. During past decades, it has become evident that the deformable cytoskeleton (CSK), an intracellular network of various filamentous biopolymers, provides a physical basis for transducing mechanical signals into biochemical responses. To understand how mechanical forces regulate cellular functions, it is necessary to first understand how the CSK develops mechanical stresses in response to applied forces, and how those stresses are propagated through the CSK where various signaling molecules are immobilized. New experimental techniques have been developed to quantify cytoskeletal mechanics, which together with new computational approaches have given rise to new theories and models for describing mechanics of living cells. In this article, we discuss current understanding of cell biomechanics by focusing on the biophysical mechanisms that are responsible for the development and transmission of mechanical stresses in the cell and their effect on cellular functions. We compare and contrast various theories and models of cytoskeletal mechanics, emphasizing common mechanisms that those theories are built upon, while not ignoring irreconcilable differences. We highlight most recent advances in the understanding of mechanotransduction in the cytoplasm of living cells and the central role of the cytoskeletal prestress in propagating mechanical forces along the cytoskeletal filaments to activate cytoplasmic enzymes. It is anticipated that advances in cell mechanics will help developing novel therapeutics to treat pulmonary diseases like asthma, pulmonary fibrosis, and chronic obstructive pulmonary disease.

Complexity of the tensegrity structure for dynamic energy and force distribution of cytoskeleton during cell spreading

Cytoskeleton plays important roles in intracellular force equilibrium and extracellular force transmission from/to attaching substrate through focal adhesions (FAs). Numerical simulations of intracellular force distribution to describe dynamic cell behaviors are still limited. The tensegrity structure comprises tension-supporting cables and compression-supporting struts that represent the actin filament and microtubule respectively, and has many features consistent with living cells. To simulate the dynamics of intracellular force distribution and total stored energy during cell spreading, the present study employed different complexities of the tensegrity structures by using octahedron tensegrity (OT) and cuboctahedron tensegrity (COT). The spreading was simulated by assigning specific connection nodes for radial displacement and attachment to substrate to form FAs. The traction force on each FA was estimated by summarizing the force carried in sounding cytoskeletal elements. The OT structure consisted of 24 cables and 6 struts and had limitations soon after the beginning of spreading by declining energy stored in struts indicating the abolishment of compression in microtubules. The COT structure, double the amount of cables and struts than the OT structure, provided sufficient spreading area and expressed similar features with documented cell behaviors. The traction force pointed inward on peripheral FAs in the spread out COT structure. The complex structure in COT provided further investigation of various FA number during different spreading stages. Before the middle phase of spreading (half of maximum spreading area), cell attachment with 8 FAs obtained minimized cytoskeletal energy. The maximum number of 12 FAs in the COT structure was required to achieve further spreading. The stored energy in actin filaments increased as cells spread out, while the energy stored in microtubules increased at initial spreading, peaked in middle phase, and then declined as cells reached maximum spreading. The dynamic flows of energy in struts imply that microtubules contribute to structure stabilization.

Nanomechanics to drive stem cells in injured tissues: insights from current research and future perspectives

Stem cells reside within tissue, ensuring its natural ability to repair an injury. They are involved in the natural repair of damaged tissue, which encompasses a complex process requiring the modulation of cell survival, extracellular matrix turnover, angiogenesis, and reverse remodeling. To date, the real reparative potential of each tissue is underestimated and noncommittal. The assessment of the biophysical properties of the extracellular environment is an innovative approach to better understand mechanisms underlying stem cell function, and consequently to develop safe and effective therapeutic strategies replacing the loss of tissue. Recent studies have focused on the role played by biomechanical signals that drive stem cell death, differentiation, and paracrinicity in a genetic and/or an epigenetic manner. Mechanical stimuli acting on the shape can influence the biochemistry and gene expression of resident stem cells and, therefore, the magnitude of biological responses that promote the healing of injured tissue. Nanotechnologies have proven to be a revolutionary tool capable of dissecting the cellular mechanosensing apparatus, allowing the intercellular cross-talk to be decoded and enabling the reparative potential of tissue to be enhanced without manipulation of stem cells. This review highlights the most relevant findings of stem cell mechanobiology and presents a fascinating perspective in regenerative medicine.

A mechanical model of actin stress fiber formation and substrate elasticity sensing in adherent cells

Tissue cells sense and respond to the stiffness of the surface on which they adhere. Precisely how cells sense surface stiffness remains an open question, though various biochemical pathways are critical for a proper stiffness response. Here, based on a simple mechanochemical model of biological friction, we propose a model for cell mechanosensation as opposed to previous more biochemically based models. Our model of adhesion complexes predicts that these cell-surface interactions provide a viscous drag that increases with the elastic modulus of the surface. The force-velocity relation of myosin II implies that myosin generates greater force when the adhesion complexes slide slowly. Then, using a simple cytoskeleton model, we show that an external force applied to the cytoskeleton causes actin filaments to aggregate and orient parallel to the direction of force application. The greater the external force, the faster this aggregation occurs. As the steady-state probability of forming these bundles reflects a balance between the time scale of bundle formation and destruction (because of actin turnover), more bundles are formed when the cytoskeleton time-scale is small (i.e., on stiff surfaces), in agreement with experiment. As these large bundles of actin, called stress fibers, appear preferentially on stiff surfaces, our mechanical model provides a mechanism for stress fiber formation and stiffness sensing in cells adhered to a compliant surface.

Contribution of the cytoskeleton to the compressive properties and recovery behavior of single cells

The cytoskeleton is known to play an important role in the biomechanical nature and structure of cells, but its particular function in compressive characteristics has not yet been fully examined. This study focused on the contribution of the main three cytoskeletal elements to the bulk compressive stiffness (as measured by the compressive modulus), volumetric or apparent compressibility changes (as further indicated by apparent Poisson's ratio), and recovery behavior of individual chondrocytes. Before mechanical testing, cytochalasin D, acrylamide, or colchicine was used to disrupt actin microfilaments, intermediate filaments, or microtubules, respectively. Cells were subjected to a range of compressive strains and allowed to recover to equilibrium. Analysis of the video recording for each mechanical event yielded relevant compressive properties and recovery characteristics related to the specific cytoskeletal disrupting agent and as a function of applied axial strain. Inhibition of actin microfilaments had the greatest effect on bulk compressive stiffness ( approximately 50% decrease compared to control). Meanwhile, intermediate filaments and microtubules were each found to play an integral role in either the diminution (compressibility) or retention (incompressibility) of original cell volume during compression. In addition, microtubule disruption had the largest effect on the "critical strain threshold" in cellular mechanical behavior (33% decrease compared to control), as well as the characteristic time for recovery ( approximately 100% increase compared to control). Elucidating the role of the cytoskeleton in the compressive biomechanical behavior of single cells is an important step toward understanding the basis of mechanotransduction and the etiology of cellular disease processes.

Mechanical and spatial determinants of cytoskeletal geodesic dome formation in cardiac fibroblasts

This study tests the hypothesis that the cell cytoskeletal (CSK) network can rearrange from geodesic dome type structures to stress fibers in response to microenvironmental cues. The CSK geodesic domes are highly organized actin microarchitectures within the cell, consisting of ordered polygonal elements. We studied primary neonatal rat cardiac fibroblasts. The cues used to trigger the interconversion between the two CSK architectures (geodesic domes and stress fibers) included factors affecting spatial order and the degree of CSK tension in the cells. Microfabricated three-dimensional substrates with micrometre sized grooves and peaks were used to alter the spatial order of cell growth in culture. CSK tension was modified by 2,3-butanedione 2-monoxime (BDM), cytochalasin D and the hyphae of Candida albicans. CSK geodesic domes occurred spontaneously in about 20% of the neonatal rat cardiac fibroblasts used in this study. Microfabricated structured surfaces produced anisotropy in the cell CSK and effectively converted geodesic domes into stress fibers in a dose-dependent manner (dependence on the period of the features). Affectors of actin structure, inhibitors of CSK tension and cell motility, e.g. BDM, cytochalasin D and the hyphae of C. albicans, suppressed or eliminated the geodesic domes. Our data suggest that the geodesic domes, similar to actin stress fibers, require maintenance of CSK integrity and tension. However, microenvironments that promote structural anisotropy in tensed cells cause the transformation of the geodesic domes into stress fibers, consistent with topographic cell guidance and some previous CSK model predictions.

Filamentous network mechanics and active contractility determine cell and tissue shape

For both cells and tissues, shape is closely correlated with function presumably via geometry-dependent distribution of tension. In this study, we identify common shape determinants spanning cell and tissue scales. For cells whose sites of adhesion are restricted to small adhesive islands on a micropatterned substrate, shape resembles a sequence of inward-curved circular arcs. The same shape is observed for fibroblast-populated collagen gels that are pinned to a flat substrate. Quantitative image analysis reveals that, in both cases, arc radii increase with the spanning distance between the pinning points. Although the Laplace law for interfaces under tension predicts circular arcs, it cannot explain the observed dependence on the spanning distance. Computer simulations and theoretical modeling demonstrate that filamentous network mechanics and contractility give rise to a modified Laplace law that quantitatively explains our experimental findings on both cell and tissue scales. Our model in conjunction with actomyosin inhibition experiments further suggests that cell shape is regulated by two different control modes related to motor contractility and structural changes in the actin cytoskeleton.

Strain hardening, avalanches, and strain softening in dense cross-linked actin networks

Actin filament networks enable the cytoskeleton to adjust to internal and external forcing. These dynamic networks can adapt to changes by dynamically adjusting their cross-links. Here, we model actin filaments as cross-linked elastic fibers of finite dimensions, with the cross-links being approximately 1 mum apart, and employ a full three-dimensional model to study their elastic properties by computer simulations. The results show compelling evidence that dense actin networks are characterized by (a) strain hardening without entropic elasticity, (b) avalanches of cross-link slippage leading to strain softening in the case of breakable cross-links, and (c) spontaneous formation of stress fibers in the case of dynamic cross-link formation and destruction.

Propagation of mechanical stress through the actin cytoskeleton toward focal adhesions: model and experiment

We investigate both theoretically and experimentally how stress is propagated through the actin cytoskeleton of adherent cells and consequentially distributed at sites of focal adhesions (FAs). The actin cytoskeleton is modeled as a two-dimensional cable network with different lattice geometries. Both prestrain, resulting from actomyosin contractility, and central application of external force, lead to finite forces at the FAs that are largely independent of the lattice geometry, but strongly depend on the exact spatial distribution of the FAs. The simulation results compare favorably with experiments with adherent fibroblasts onto which lateral force is exerted using a microfabricated pillar. For elliptical cells, central application of external force along the long axis leads to two large stress regions located obliquely opposite to the pulling direction. For elliptical cells pulled along the short axis as well as for circular cells, there is only one region of large stress opposite to the direction of pull. If in the computer simulations FAs are allowed to rupture under force for elliptically elongated and circular cell shapes, then morphologies arise which are typical for migrating fibroblasts and keratocytes, respectively. The same effect can be obtained also by internally generated force, suggesting a mechanism by which cells can control their migration morphologies.

Cell and biomolecular mechanics in silico

Recent developments in computational cell and biomolecular mechanics have provided valuable insights into the mechanical properties of cells, subcellular components and biomolecules, while simultaneously complementing new experimental techniques used for deciphering the structure-function paradigm in living cells. These computational approaches have direct implications in understanding the state of human health and the progress of disease and can therefore aid immensely in the diagnosis and treatment of diseases. We provide an overview of the computational approaches that are currently used in understanding various aspects of cell and bimolecular mechanics. Our emphasis is on state-of-the-art techniques and the progress made in addressing key challenges in biomechanics.

Tools to study cell mechanics and mechanotransduction

Analysis of how cells sense and respond to mechanical stress has been limited by the availability of techniques that can apply controlled mechanical forces to living cells while simultaneously measuring changes in cell and molecular distortion, as well as alterations of intracellular biochemistry. We have confronted this challenge by developing new engineering methods to measure and manipulate the mechanical properties of cells and their internal cytoskeletal and nuclear frameworks, and by combining them with molecular cell biological techniques that rely on microscopic analysis and real-time optical readouts of biochemical signaling. In this chapter, we describe techniques like microcontact printing, magnetic twisting cytometry, and magnetic pulling cytometry that can be systematically used to study the molecular basis of cellular mechanotransduction.

Divided medium-based model for analyzing the dynamic reorganization of the cytoskeleton during cell deformation

Cell deformability and mechanical responses of living cells depend closely on the dynamic changes in the structural architecture of the cytoskeleton (CSK). To describe the dynamic reorganization and the heterogeneity of the prestressed multi-modular CSK, we developed a two-dimensional model for the CSK which was taken to be a system of tension and compression interactions between the nodes in a divided medium. The model gives the dynamic reorganization of the CSK consisting of fast changes in connectivity between nodes during medium deformation and the resulting mechanical behavior is consistent with the strain-hardening and prestress-induced stiffening observed in cells in vitro. In addition, the interaction force networks which occur and balance to each other in the model can serve to identify the main CSK substructures: cortex, stress fibers, intermediate filaments, microfilaments, microtubules and focal adhesions. Removing any of these substructures results in a loss of integrity in the model and a decrease in the prestress and stiffness, and suggests that the CSK substructures are highly interdependent. The present model may therefore provide a useful tool for understanding the cellular processes involving CSK reorganization, such as mechanotransduction, migration and adhesion processes.

Measurement and characterization of whole-cell mechanical behavior

An understanding of whole-cell mechanical behavior can provide insight into cellular responses to mechanical loading and diseases in which such responses are altered. However, this aspect of cellular mechanical behavior has received limited attention. In this study, we used the atomic force microscope (AFM) in conjunction with several mechanical characterization methods (Hertz contact theory, an exponential equation, and a parallel-spring recruitment model) to establish a mechanically rigorous method for measuring and characterizing whole-cell mechanical behavior in the deformation range 0-500 nm. Using MC3T3-E1 osteoblasts, measurement repeatability was assessed by performing multiple loading cycles on individual cells. Despite variability in measurements, repeatability of the measurement technique was statistically confirmed. The measurement technique also proved acceptable since only 5% of the total variance across all measurements was due to variations within measurements for a single cell. The parallel-spring recruitment model, a single-parameter model, accurately described the measured nonlinear force-deformation response (R2>0.99) while providing a mechanistic explanation of whole-cell mechanical behavior. Taken together, the results should improve the capabilities of the AFM to probe whole-cell mechanical behavior. In addition, the success of the parallel-spring recruitment model provides insight into the micromechanical basis of whole-cell behavior.

The effects of morphology, confluency, and phenotype on whole-cell mechanical behavior

Emerging evidence indicates that cellular mechanical behavior can be altered by disease, drug treatment, and mechanical loading. To effectively investigate how disease and mechanical or biochemical treatments influence cellular mechanical behavior, it is imperative to determine the source of large inter-cell differences in whole-cell mechanical behavior within a single cell line. In this study, we used the atomic force microscope to investigate the effects of cell morphological parameters and confluency on whole-cell mechanical behavior for osteoblastic and fibroblastic cells. For nonconfluent cells, projected nucleus area, cell area, and cell aspect ratio were not correlated with mechanical behavior (p>or=0.46), as characterized by a parallel-spring recruitment model. However, measured force-deformation responses were statistically different between osteoblastic and fibroblastic cells (p<0.001) and between confluent and nonconfluent cells (p<0.001). Osteoblastic cells were 2.3-2.8 times stiffer than fibroblastic cells, and confluent cells were 1.5-1.8 times stiffer than nonconfluent cells. The results indicate that structural differences related to phenotype and confluency affect whole-cell mechanical behavior, while structural differences related to global morphology do not. This suggests that cytoskeleton structural parameters, such as filament density, filament crosslinking, and cell-cell and cell-matrix attachments, dominate inter-cell variability in whole-cell mechanical behavior.