A multi-structural single cell model of force-induced interactions of cytoskeletal components

Discrete network models of endothelial cells and their interactions with the substrate

Endothelial cell monolayers line the inner surfaces of blood and lymphatic vessels. They are continuously exposed to different mechanical loads, which may trigger mechanobiological signals and hence play a role in both physiological and pathological processes. Computer-based mechanical models of cells contribute to a better understanding of the relation between cell-scale loads and cues and the mechanical state of the hosting tissue. However, the confluency of the endothelial monolayer complicates these approaches since the intercellular cross-talk needs to be accounted for in addition to the cytoskeletal mechanics of the individual cells themselves. As a consequence, the computational approach must be able to efficiently model a large number of cells and their interaction. Here, we simulate cytoskeletal mechanics by means of molecular dynamics software, generally suitable to deal with large, locally interacting systems. Methods were developed to generate models of single cells and large monolayers with hundreds of cells. The single-cell model was considered for a comparison with experimental data. To this end, we simulated cell interactions with a continuous, deformable substrate, and computationally replicated multistep traction force microscopy experiments on endothelial cells. The results indicate that cell discrete network models are able to capture relevant features of the mechanical behaviour and are thus well-suited to investigate the mechanics of the large cytoskeletal network of individual cells and cell monolayers.

Analysis of the Adherent Cell Response to the Substrate Stiffness Using Tensegrity

Cell's shape is dependent on the cytoskeleton mechanical properties. Hybrid models were developed that combine the discrete structure for the cytoskeleton and continuum parts for other cell organelles. Tensegrity-based structures that consist of tensile and compression elements are useful models to understand the cytoskeleton mechanical behavior. In this study, we are looking to examine the reaction of the cell to a variety of substrate stiffnesses and explain the relationship between cell behavior and substrate mechanical properties. However, which tensegrity structure is appropriate for modeling a living cell? Is the structure's complexity play a major role? We used two spherical tensegrities with different complexities to assess the impact of the structure on the cell's mechanical response versus substrate's stiffness. Six- and twelve-strut tensegrities together with membrane, cytoplasm, nucleoskeleton, and nucleus envelope were assembled in Abaqus package to create a hybrid cell model. A compressive load was applied to the cell model and the reaction forces versus deflection curves were analyzed for number of substrate stiffness values. By analyzing the difference due to two different tensegrities it became clear that the lower density structure is a better choice for modeling stiffer cells. It was also found that the six-strut tensegrity is sensitive to higher range of substrate stiffness.

Macrophages modulate stiffness-related foreign body responses through plasma membrane deformation

Implants are widely used in medical applications and yet macrophage-mediated foreign body reactions caused by implants severely impact their therapeutic effects. Although the extensive use of multiple surface modifications has been introduced to provide some mitigation of fibrosis, little is known about how macrophages recognize the stiffness of the implant and thus influence cell behaviors. Here, we demonstrated that macrophage stiffness sensing leads to differential inflammatory activation, resulting in different degrees of fibrosis. The potential mechanism for macrophage stiffness sensing in the early adhesion stages tends to involve cell membrane deformations on substrates with different stiffnesses. Combining theory and experiments, we show that macrophages exert traction stress on the substrate through adhesion and altered membrane curvature, leading to the uneven distribution of the curvature-sensing protein Baiap2, resulting in cytoskeleton remodeling and inflammation inhibition. This study introduces a physical model feedback mechanism for early cellular stiffness sensing based on cell membrane deformation, offering perspectives for future material design and targeted therapies.

A high-throughput microfluidic device inspired by the Wheatstone bridge principle for characterizing the mechanical properties of single cells

The mechanical properties of single cells have been recognized as biomarkers for identifying individual cells and diagnosing human diseases. Microfluidic devices based on the flow cytometry principle, which are not limited by the vision field of a microscope and can achieve a very high throughput, have been extensively adopted to measure the mechanical properties of single cells. However, these kinds of microfluidic devices usually required pressure-driven pumps with a very low flow rate and high precision. In this study, we developed a high-throughput microfluidic device inspired by the Wheatstone bridge principle for characterizing the mechanical properties of single cells. The microfluidic analogue of the Wheatstone bridge not only took advantage of flow cytometry, but also allowed precise control of a very low flow rate through the constricted channel with a higher input flow rate generated by a commercially available pressure-driven pump. Under different input flow rates of the pump, the apparent elastic moduli and the fluidity of osteosarcoma (U-2OS) cells and cervical carcinoma (HeLa) cells were measured by monitoring their dynamic deformations passing through the bridge-channel with different sizes of rectangular constrictions. The results showed that the input flow rate had little effect on measuring the mechanical properties of the cells, while the ratio of cell radius to effective constriction radius was different, i.e., for U-2OS cells it was 1.20 and for HeLa cells it was 1.09. Under this condition compared with predecessors, our statistic results of cell mechanical properties exhibited minimal errors. Furthermore, the cell viability after measurements was kept above 90% that demonstrated the non-destructive property of our proposed method.

The role of cellular traction forces in deciphering nuclear mechanics

Cellular forces exerted on the extracellular matrix (ECM) during adhesion and migration under physiological and pathological conditions regulate not only the overall cell morphology but also nuclear deformation. Nuclear deformation can alter gene expression, integrity of the nuclear envelope, nucleus-cytoskeletal connection, chromatin architecture, and, in some cases, DNA damage responses. Although nuclear deformation is caused by the transfer of forces from the ECM to the nucleus, the role of intracellular organelles in force transfer remains unclear and a challenging area of study. To elucidate nuclear mechanics, various factors such as appropriate biomaterial properties, processing route, cellular force measurement technique, and micromanipulation of nuclear forces must be understood. In the initial phase of this review, we focused on various engineered biomaterials (natural and synthetic extracellular matrices) and their manufacturing routes along with the properties required to mimic the tumor microenvironment. Furthermore, we discussed the principle of tools used to measure the cellular traction force generated during cell adhesion and migration, followed by recently developed techniques to gauge nuclear mechanics. In the last phase of this review, we outlined the principle of traction force microscopy (TFM), challenges in the remodeling of traction forces, microbead displacement tracking algorithm, data transformation from bead movement, and extension of 2-dimensional TFM to multiscale TFM.

Impact of physiological loads of arterial wall on nucleus deformation in endothelial cells: A computational study

INTRODUCTION

Computational modeling can enhance the understanding of cell mechanics. To achieve this, finite element models of endothelial cells were proposed with shapes mimicking their natural state inside the endothelium within the cardiovascular system. Implementing the recently proposed bendo-tensegrity concept, these models consider flexural (buckling) as well as tensional/compressional behavior of microtubules and also incorporate the waviness of intermediate filaments.

MATERIALS AND METHODS

Four different models were created (flat and domed hexagons, both regular and elongated in the direction of blood flow) and loaded by biaxial deformation, blood pressure, and shear load from blood flow - natural physiological conditions of the arterial endothelium - aiming to investigate the "in situ" mechanical response of the cell.

RESULTS

The impact of individual components of loads on the nucleus deformation (more specifically on the first principal strain) potentially influencing mechanotransduction was investigated and the role of the cytoskeleton and its constituents in the mechanical response of the endothelial cell was assessed. The results show (i) the impact of pulsating blood pressure on cyclic deformations of the nucleus, which increase substantially with decreasing axial pre-stretch of the cell, (ii) the importance of relatively low shear stresses in the cell response and nucleus deformation.

CONCLUSION

Not only the pulsatile blood pressure but also the wall shear stress may induce significant deformation of the nucleus and thus trigger remodelation processes in endothelial cells.

Simultaneous Measurement of Single-Cell Mechanics and Cell-to-Materials Adhesion Using Fluidic Force Microscopy

The connection between cells and their substrate is essential for biological processes such as cell migration. Atomic force microscopy nanoindentation has often been adopted to measure single-cell mechanics. Very recently, fluidic force microscopy has been developed to enable rapid measurements of cell adhesion. However, simultaneous characterization of the cell-to-material adhesion and viscoelastic properties of the same cell is challenging. In this study, we present a new approach to simultaneously determine these properties for single cells, using fluidic force microscopy. For MCF-7 cells grown on tissue-culture-treated polystyrene surfaces, we found that the adhesive force and adhesion energy were correlated for each cell. Well-spread cells tended to have stronger adhesion, which may be due to the greater area of the contact between cellular adhesion receptors and the surface. By contrast, the viscoelastic properties of MCF-7 cells cultured on the same surface appeared to have little dependence on cell shape. This methodology provides an integrated approach to better understand the biophysics of multiple cell types.

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.

Finite element analysis of the influence of cyclic strain on cells anchored to substrates with varying properties

The response of cytoskeleton to mechanical cues plays a pivotal role in understanding several aspects of cellular growth, migration, and cell-cell and cell-matrix interactions under normal and diseased conditions. Finite element analysis (FEA) has become a powerful computational technique to study the response of cytoskeleton in the maintenance of overall cellular mechanics. With the revelation of role of external mechanical microenvironment on cell mechanics, FEA models have also been developed to simulate the effect of substrate stiffness on the mechanical properties of cancer cells. However, the models developed so far model cellular response under static mode, whereas in physiological condition, cells always experience dynamic loading conditions. To develop a more accurate model of cell-extracellular matrix (ECM) interactions, this paper models the cytoskeleton and other parts of the cell by beam and solid elements respectively, assuming spherical morphology of the cell. The stiffness and roughness of extracellular matrix were varied. Furthermore, static and dynamic sinusoidal loads were applied through a flat plate indenter on the cell along with providing sinusoidal strain at the substrate. It is observed that due to axial loading, cell reaches a plastic region, and when the sinusoidal loading is added to the axial load, the cell experiences permanent deformation. Degradation of the cytoskeleton elements and a physiologically more relevant spherical cap shape of the cell were also considered during the analysis. This study suggests that asperity topology of the substrate and indirect cyclic load can play a significant role in the shape alterations and motion of a cell.

Auxeticity in biosystems: an exemplification of its effects on the mechanobiology of heterogeneous living cells

Auxeticity (negative Poisson's ratio) is the unique mechanical property found in an extensive variety of materials, such as metals, graphene, composites, polymers, foams, fibers, ceramics, zeolites, silicates and biological tissues. The enhanced mechanical features of the auxetic materials have motivated scientists to design, engineer and manufacture man-made auxetic materials to fully leverage their capabilities in different fields of research applications, including aeronautics, medical, protective equipments, smart sensors, filter cleaning, and so on. Atomic force microscopy (AFM) indentation is one of the most widely used methods for characterizing the mechanical properties and response of the living cells. In this contribution, we highlight main consequences of auxeticity for biosystems and provide a representative example to quantify the effect of nucleus auxeticity on the force response of the embryonic stem cells. A parametric study has been conducted on a heterogeneous stem cell to evaluate the effect of nucleus diameter, nucleus elasticity, indenter's shape and location on the force-indentation curve. The developed model has also been validated with the recently reported experimental studies available in the literature. Our results suggest that the nucleus auxeticity plays a profound role in cell mechanics especially for large size nucleus. We also report the mechanical stresses induced within the hyperelastic cell model under different loading conditions that would be quite useful in decoding the interrelations between mechanical stimuli and cellular behavior of auxetic biosystems. Finally, current and potential areas of applications of our findings for regenerative therapies, tissue engineering, 3 D/4D bioprinting, and the development of meta-biomaterials are discussed.

In silico stress fibre content affects peak strain in cytoplasm and nucleus but not in the membrane for uniaxial substrate stretch

Existing in silico models for single cell mechanics feature limited representations of cytoskeletal structures that contribute substantially to the mechanics of a cell. We propose a micromechanical hierarchical approach to capture the mechanical contribution of actin stress fibres. For a cell-specific fibroblast geometry with membrane, cytoplasm and nucleus, the Mori-Tanaka homogenization method was employed to describe cytoplasmic inhomogeneities and constitutive contribution of actin stress fibres. The homogenization was implemented in a finite element model of the fibroblast attached to a substrate through focal adhesions. Strain in cell membrane, cytoplasm and nucleus due to uniaxial substrate stretch was assessed for different stress fibre volume fractions and different elastic modulus of the substrate. A considerable decrease of the peak strain with increasing stress fibre content was observed in cytoplasm and nucleus but not the membrane, whereas the peak strain in cytoplasm, nucleus and membrane increased for increasing elastic modulus of the substrate. Finite element mesh of reconstructed human fibroblast and intracellular strain distribution in cell subjected to substrate stretch.

Finite element study of stem cells under fluid flow for mechanoregulation toward osteochondral cells

Investigating the effects of mechanical stimuli on stem cells under in vitro and in vivo conditions is a very important issue to reach better control on cellular responses like growth, proliferation, and differentiation. In this regard, studying the effects of scaffold geometry, steady, and transient fluid flow, as well as influence of different locations of the cells lodged on the scaffold on effective mechanical stimulations of the stem cells are of the main goals of this study. For this purpose, collagen-based scaffolds and implicit surfaces of the pore architecture was used. In this study, computational fluid dynamics and fluid-structure interaction method was used for the computational simulation. The results showed that the scaffold microstructure and the pore architecture had an essential effect on accessibility of the fluid to different portions of the scaffold. This leads to the optimization of shear stress and hydrodynamic pressure in different surfaces of the scaffold for better transportation of oxygen and growth factors as well as for optimized mechanoregulative responses of cell-scaffold interactions. Furthermore, the results indicated that the HP scaffold provides more optimizer surfaces to culture stem cells rather than Gyroid and IWP scaffolds. The results of exerting oscillatory fluid flow into the HP scaffold showed that the whole surface of the HP scaffold expose to the shear stress between 0.1 and 40 mPa and hydrodynamics factors on the scaffold was uniform. The results of this study could be used as an aid for experimentalists to choose optimist fluid flow conditions and suitable situation for cell culture.

Spatial correlation of cell stiffness and traction forces in cancer cells measured with combined SICM and TFM

The mechanical properties of cancer cells at the single-cell and the subcellular level might be the key for answering long-standing questions in the diagnosis and treatment of cancer. However, the subcellular distribution of two main mechanical properties, cell stiffness and traction forces, has been investigated only rarely and qualitatively yet. Here, we present the first direct combination of scanning ion conductance microscopy (SICM) and traction force microscopy (TFM), which we used to identify a correlation between the local stiffness and the local traction force density in living cells. We found a correlation in normal breast epithelial cells, but no correlation in cancerous breast epithelial cells. This indicates that the interplay between cell stiffness and traction forces is altered in cancer cells as compared to healthy cells, which might give new insight in the research field of cancer cell mechanobiology.

Finite Element Simulations of Mechanical Behaviour of Endothelial Cells

Biomechanical models based on the finite element method have already shown their potential in the simulation of the mechanical behaviour of cells. For instance, development of atherosclerosis is accelerated by damage of the endothelium, a monolayer of endothelial cells on the inner surface of arteries. Finite element models enable us to investigate mechanical factors not only at the level of the arterial wall but also at the level of individual cells. To achieve this, several finite element models of endothelial cells with different shapes are presented in this paper. Implementing the recently proposed bendotensegrity concept, these models consider the flexural behaviour of microtubules and incorporate also waviness of intermediate filaments. The suspended and adherent cell models are validated by comparison of their simulated force-deformation curves with experiments from the literature. The flat and dome cell models, mimicking natural cell shapes inside the endothelial layer, are then used to simulate their response in compression and shear which represent typical loads in a vascular wall. The models enable us to analyse the role of individual cytoskeletal components in the mechanical responses, as well as to quantify the nucleus deformation which is hypothesized to be the quantity decisive for mechanotransduction.

Evaluation of quasi-static and dynamic nanomechanical properties of bone-metastatic breast cancer cells using a nanoclay cancer testbed

In recent years, there has been increasing interest in investigating the mechanical properties of individual cells to delineate disease mechanisms. Reorganization of cytoskeleton facilitates the colonization of metastatic breast cancer at bone marrow space, leading to bone metastasis. Here, we report evaluation of mechanical properties of two breast cancer cells with different metastatic ability at the site of bone metastases, using quasi-static and dynamic nanoindentation methods. Our results showed that the significant reduction in elastic modulus along with increased liquid-like behavior of bone metastasized MCF-7 cells was induced by depolymerization and reorganization of F-actin to the adherens junctions, whereas bone metastasized MDA-MB-231 cells showed insignificant changes in elastic modulus and F-actin reorganization over time, compared to their respective as-received counterparts. Taken together, our data demonstrate evolution of breast cancer cell mechanics at bone metastases.

Myosin light chain kinase mediates intestinal barrier dysfunction following simulated microgravity based on proteomic strategy

Microgravity induces injury of intestinal barrier. However, the underlying mechanism remains unclear. The present study aimed to investigate the pathological change of intestinal mucosa induced by long term simulated microgravity and to explore its etiological mechanism using a proteomic approach. The well accepted tail-suspended rat model was used to simulate microgravity. The damage of rat small intestine was evaluated via histological and molecular test, and a label-free comparative proteomic strategy was used to determine the molecular mechanism. Simulated microgravity for 21 days damaged intestine barrier with decreased numbers of the goblet cells, large intercellular space, and down-regulated adhesion molecules, accompanied by increased intestinal permeability. Proteomic analysis identified 416 differentially expressed proteins and showed simulated microgravity dramatically down-regulated the adhesion molecules and deteriorated several pathways for metabolism, focal adhesion, and regulation of actin cytoskeleton. Western-blot analysis confirmed that myosin regulatory light chain (MLC) 12B was significantly down-regulated, while rho-associated protein kinase, myosin light chain kinase (MLCK), and phosphorylated MLC were dramatically up-regulated. Taken together, these data reveal that down-regulation of adhesion molecules and MLCK dependent up-regulation MLC phosphorylation mediate intestinal barrier dysfunction during simulated microgravity injury. Our results also indicate that regulation of epithelial MLCK is a potential target for the therapeutic treatment of microgravity injury.

Mechanics of the cell: Interaction mechanisms and mechanobiological models

The importance of cell mechanics has long been recognized for the cell development and function. Biomechanics plays an important role in cell metabolism, regulation of mechanotransduction pathways and also modulation of nuclear response. The mechanical properties of the cell are likely determined by, among many others, the cytoskeleton elasticity, membrane tension and cell-substrate adhesion. This coordinated but complex mechanical interplay is required however, for the cell to respond to and influence in a reciprocal manner the chemical and mechanical signals from the extracellular matrix (ECM). In an effort to better and more fully understand the cell mechanics, the role of nuclear mechanics has emerged as an important contributor to the overall cellular mechanics. It is not too difficult to appreciate the physical connection between the nucleus and the cytoskeleton network that may be connected to the ECM through the cell membrane. Transmission of forces from ECM through this connection is essential for a wide range of cellular behaviors and functions such as cytoskeletal reorganization, nuclear movement, cell migration and differentiation. Unlike the cellular mechanics that can be measured using a number of biophysical techniques that were developed in the past few decades, it still remains a daunting challenge to probe the nuclear mechanics directly. In this paper, we therefore aim to provide informative description of the cell membrane and cytoskeleton mechanics, followed by unique computational modeling efforts to elucidate the nucleus-cytoskeleton coupling. Advances in our knowledge of complete cellular biomechanics and mechanotransduction may lead to clinical relevance and applications in mechano-diseases such as atherosclerosis, stem cell-based therapies, and the development of tissue engineered products.

Nonlinear Elastic and Inelastic Properties of Cells

Mechanical forces play an important role in various physiological processes, such as morphogenesis, cytokinesis, and migration. Thus, in order to illuminate mechanisms underlying these physiological processes, it is crucial to understand how cells deform and respond to external mechanical stimuli. During recent decades, the mechanical properties of cells have been studied extensively using diverse measurement techniques. A number of experimental studies have shown that cells are far from linear elastic materials. Cells exhibit a wide variety of nonlinear elastic and inelastic properties. Such complicated properties of cells are known to emerge from unique mechanical characteristics of cellular components. In this review, we introduce major cellular components that largely govern cell mechanical properties and provide brief explanations of several experimental techniques used for rheological measurements of cell mechanics. Then, we discuss the representative nonlinear elastic and inelastic properties of cells. Finally, continuum and discrete computational models of cell mechanics, which model both nonlinear elastic and inelastic properties of cells, will be described.

Stress fiber anisotropy contributes to force-mode dependent chromatin stretching and gene upregulation in living cells

Living cells and tissues experience various complex modes of forces that are important in physiology and disease. However, how different force modes impact gene expression is elusive. Here we apply local forces of different modes via a magnetic bead bound to the integrins on a cell and quantified cell stiffness, chromatin deformation, and DHFR (dihydrofolate reductase) gene transcription. In-plane stresses result in lower cell stiffness than out-of-plane stresses that lead to bead rolling along the cell long axis (i.e., alignment of actin stress fibers) or at different angles (90° or 45°). However, chromatin stretching and ensuing DHFR gene upregulation by the in-plane mode are similar to those induced by the 45° stress mode. Disrupting stress fibers abolishes differences in cell stiffness, chromatin stretching, and DHFR gene upregulation under different force modes and inhibiting myosin II decreases cell stiffness, chromatin deformation, and gene upregulation. Theoretical modeling using discrete anisotropic stress fibers recapitulates experimental results and reveals underlying mechanisms of force-mode dependence. Our findings suggest that forces impact biological responses of living cells such as gene transcription via previously underappreciated means.

Living cells and tissues experience various complex modes of forces but how different force modes impact gene expression is elusive. Here authors apply forces via magnetic beads to integrins on a cell surface and observe force-mode dependent chromatin stretching and gene upregulation in cells and identify underlying mechanisms.

A NanoFE simulation-based surrogate machine learning model to predict mechanical functionality of protein networks from live confocal imaging

Sub-cellular mechanics plays a crucial role in a variety of biological functions and dysfunctions. Due to the strong structure-function relationship in cytoskeletal protein networks, light can be shed on their mechanical functionality by investigating their structures. Here, we present a data-driven approach employing a combination of confocal live imaging of fluorescent tagged protein networks, in silico mechanical experiments and machine learning to investigate this relationship. Our designed image processing and nanoFE mechanical simulation framework resolves the structure and mechanical behaviour of cytoskeletal networks and the developed gradient boosting surrogate models linking network structure to its functionality. In this study, for the first time, we perform mechanical simulations of Filamentous Temperature Sensitive Z (FtsZ) complex protein networks with realistic network geometry depicting its skeletal functionality inside organelles (here, chloroplasts) of the moss Physcomitrella patens. Training on synthetically produced simulation data enables predicting the mechanical characteristics of FtsZ network purely based on its structural features (R2⩾0.93), therefore allowing to extract structural principles enabling specific mechanical traits of FtsZ, such as load bearing and resistance to buckling failure in case of large network deformation.

Recent Advances on the Model, Measurement Technique, and Application of Single Cell Mechanics

Since the cell was discovered by humans, it has been an important research subject for researchers. The mechanical response of cells to external stimuli and the biomechanical response inside cells are of great significance for maintaining the life activities of cells. These biomechanical behaviors have wide applications in the fields of disease research and micromanipulation. In order to study the mechanical behavior of single cells, various cell mechanics models have been proposed. In addition, the measurement technologies of single cells have been greatly developed. These models, combined with experimental techniques, can effectively explain the biomechanical behavior and reaction mechanism of cells. In this review, we first introduce the basic concept and biomechanical background of cells, then summarize the research progress of internal force models and experimental techniques in the field of cell mechanics and discuss the latest mechanical models and experimental methods. We summarize the application directions of cell mechanics and put forward the future perspectives of a cell mechanics model.

Biological cells and coupled electro-mechanical effects: The role of organelles, microtubules, and nonlocal contributions

Biological cells are exposed to a variety of mechanical loads throughout their life cycles that eventually play an important role in a wide range of cellular processes. The understanding of cell mechanics under the application of external stimuli is important for capturing the nuances of physiological and pathological events. Such critical knowledge will play an increasingly vital role in modern medical therapies such as tissue engineering and regenerative medicine, as well as in the development of new remedial treatments. At present, it is well known that the biological molecules exhibit piezoelectric properties that are of great interest for medical applications ranging from sensing to surgery. In the current study, a coupled electro-mechanical model of a biological cell has been developed to better understand the complex behaviour of biological cells subjected to piezoelectric and flexoelectric properties of their constituent organelles under the application of external forces. Importantly, a more accurate modelling paradigm has been presented to capture the nonlocal flexoelectric effect in addition to the linear piezoelectric effect based on the finite element method. Major cellular organelles considered in the developed computational model of the biological cell are the nucleus, mitochondria, microtubules, cell membrane and cytoplasm. The effects of variations in the applied forces on the intrinsic piezoelectric and flexoelectric contributions to the electro-elastic response have been systematically investigated along with accounting for the variation in the coupling coefficients. In addition, the effect of mechanical degradation of the cytoskeleton on the electro-elastic response has also been quantified. The present studies suggest that flexoelectricity could be a dominant electro-elastic coupling phenomenon, exhibiting electric fields that are four orders of magnitude higher than those generated by piezoelectric effects alone. Further, the output of the coupled electro-mechanical model is significantly dependent on the variation of flexoelectric coefficients. We have found that the mechanical degradation of the cytoskeleton results in the enhancement of both the piezo and flexoelectric responses associated with electro-mechanical coupling. In general, our study provides a framework for more accurate quantification of the mechanical/electrical transduction within the biological cells that can be critical for capturing the complex mechanisms at cellular length scales.

A dynamical model of oncotripsy by mechanical cell fatigue: selective cancer cell ablation by low-intensity pulsed ultrasound

The method of oncotripsy, first proposed in Heyden & Ortiz (Heyden & Ortiz 2016 J. Mech. Phys. Solids 92, 164-175 (doi:10.1016/j.jmps.2016.04.016)), exploits aberrations in the material properties and morphology of cancerous cells in order to ablate them selectively by means of tuned low-intensity pulsed ultrasound. We propose the dynamical model of oncotripsy that follows as an application of cell dynamics, statistical mechanical theory of network elasticity and 'birth-death' kinetics to describe the processes of damage and repair of the cytoskeleton. We also develop a reduced dynamical model that approximates the three-dimensional dynamics of the cell and facilitates parametric studies, including sensitivity analysis and process optimization. We show that the dynamical model predicts-and provides a conceptual basis for understanding-the oncotripsy effect and other trends in the data of Mittelstein et al. (Mittelstein et al. 2019 Appl. Phys. Lett. 116, 013701 (doi:10.1063/1.5128627)), for cells in suspension, including the dependence of cell-death curves on cell and process parameters.

A dynamical model of oncotripsy by mechanical cell fatigue: selective cancer cell ablation by low-intensity pulsed ultrasound

The method of oncotripsy, first proposed in Heyden & Ortiz (Heyden & Ortiz 2016 J. Mech. Phys. Solids 92, 164-175 (doi:10.1016/j.jmps.2016.04.016)), exploits aberrations in the material properties and morphology of cancerous cells in order to ablate them selectively by means of tuned low-intensity pulsed ultrasound. We propose the dynamical model of oncotripsy that follows as an application of cell dynamics, statistical mechanical theory of network elasticity and 'birth-death' kinetics to describe the processes of damage and repair of the cytoskeleton. We also develop a reduced dynamical model that approximates the three-dimensional dynamics of the cell and facilitates parametric studies, including sensitivity analysis and process optimization. We show that the dynamical model predicts-and provides a conceptual basis for understanding-the oncotripsy effect and other trends in the data of Mittelstein et al. (Mittelstein et al. 2019 Appl. Phys. Lett. 116, 013701 (doi:10.1063/1.5128627)), for cells in suspension, including the dependence of cell-death curves on cell and process parameters.

Sensing and Modelling Mechanical Response in Large Deformation Indentation of Adherent Cell Using Atomic Force Microscopy

The mechanical behaviour of adherent cells when subjected to the local indentation can be modelled via various approaches. Specifically, the tensegrity structure has been widely used in describing the organization of discrete intracellular cytoskeletal components, including microtubules (MTs) and microfilaments. The establishment of a tensegrity model for adherent cells has generally been done empirically, without a mathematically demonstrated methodology. In this study, a rotationally symmetric prism-shaped tensegrity structure is introduced, and it forms the basis of the proposed multi-level tensegrity model. The modelling approach utilizes the force density method to mathematically assure self-equilibrium. The proposed multi-level tensegrity model was developed by densely distributing the fundamental tensegrity structure in the intracellular space. In order to characterize the mechanical behaviour of the adherent cell during the atomic force microscopy (AFM) indentation with large deformation, an integrated model coupling the multi-level tensegrity model with a hyperelastic model was also established and applied. The coefficient of determination between the computational force-distance (F-D) curve and the experimental F-D curve was found to be at 0.977 in the integrated model on average. In the simulation range, along with the increase in the overall deformation, the local stiffness contributed by the cytoskeletal components decreased from 75% to 45%, while the contribution from the hyperelastic components increased correspondingly.

Revealing hidden information in osteoblast's mechanotransduction through analysis of time patterns of critical events

BACKGROUND

Mechanotransduction in bone cells plays a pivotal role in osteoblast differentiation and bone remodelling. Mechanotransduction provides the link between modulation of the extracellular matrix by mechanical load and intracellular activity. By controlling the balance between the intracellular and extracellular domains, mechanotransduction determines the optimum functionality of skeletal dynamics. Failure of this relationship was suggested to contribute to bone-related diseases such as osteoporosis.

RESULTS

A hybrid mechanical and agent-based model (Mech-ABM), simulating mechanotransduction in a single osteoblast under external mechanical perturbations, was utilised to simulate and examine modulation of the activation dynamics of molecules within mechanotransduction on the cellular response to mechanical stimulation. The number of molecules and their fluctuations have been analysed in terms of recurrences of critical events. A numerical approach has been developed to invert subordination processes and to extract the direction processes from the molecular signals in order to derive the distribution of recurring events. These predict that there are large fluctuations enclosing information hidden in the noise which is beyond the dynamic variations of molecular baselines. Moreover, studying the system under different mechanical load regimes and altered dynamics of feedback loops, illustrate that the waiting time distributions of each molecule are a signature of the system's state.

CONCLUSIONS

The behaviours of the molecular waiting times change with the changing of mechanical load regimes and altered dynamics of feedback loops, presenting the same variation of patterns for similar interacting molecules and identifying specific alterations for key molecules in mechanotransduction. This methodology could be used to provide a new tool to identify potent molecular candidates to modulate mechanotransduction, hence accelerate drug discovery towards therapeutic targets for bone mass upregulation.

Localized Axolemma Deformations Suggest Mechanoporation as Axonal Injury Trigger

Traumatic brain injuries are a leading cause of morbidity and mortality worldwide. With almost 50% of traumatic brain injuries being related to axonal damage, understanding the nature of cellular level impairment is crucial. Experimental observations have so far led to the formulation of conflicting theories regarding the cellular primary injury mechanism. Disruption of the axolemma, or alternatively cytoskeletal damage has been suggested mainly as injury trigger. However, mechanoporation thresholds of generic membranes seem not to overlap with the axonal injury deformation range and microtubules appear too stiff and too weakly connected to undergo mechanical breaking. Here, we aim to shed a light on the mechanism of primary axonal injury, bridging finite element and molecular dynamics simulations. Despite the necessary level of approximation, our models can accurately describe the mechanical behavior of the unmyelinated axon and its membrane. More importantly, they give access to quantities that would be inaccessible with an experimental approach. We show that in a typical injury scenario, the axonal cortex sustains deformations large enough to entail pore formation in the adjoining lipid bilayer. The observed axonal deformation of 10–12% agree well with the thresholds proposed in the literature for axonal injury and, above all, allow us to provide quantitative evidences that do not exclude pore formation in the membrane as a result of trauma. Our findings bring to an increased knowledge of axonal injury mechanism that will have positive implications for the prevention and treatment of brain injuries.

An iterative finite element-based method for solving inverse problems in traction force microscopy

BACKGROUND AND OBJECTIVE

During the last years different model solutions were proposed for solving cell forces under different conditions. The solution relies on a deformation field that is obtained under cell relaxation with a chemical cocktail. Once the deformation field of the matrix is determined, cell forces can be computed by an inverse algorithm, given the mechanical properties of the matrix. Most of the Traction Force Microscopy (TFM) methods presented so far relied on a linear stress-strain response of the matrix. However, the mechanical response of some biopolymer networks, such as collagen gels is more complex. In this work, we present a numerical method for solving cell forces on non-linear materials.

METHODS

The proposed method relies on solving the inverse problem based on an iterative optimization. The objective function is defined by least-square minimization of the difference between the target and the current computed deformed configuration of the cell, and the iterative formulation is based on the solution of several direct mechanical problems. The model presents a well-posed discretized inverse elasticity problem in the absence of regularization. The algorithm can be easily implemented in any kind of Finite Element (FE) code as a sequence of different standard FE analysis.

RESULTS

To illustrate the proposed iterative formulation we apply the theoretical model to some illustrative examples by using real experimental data of Normal Human Dermal Fibroblast cells (NHDF) migrating inside a 2 mg/ml collagen-based gel. Different examples of application have been simulated to test the inverse numerical model proposed and to investigate the effect of introducing the correct cell properties onto the obtained cell forces. The algorithm converges after a small number of iterations, generating errors of around 5% for the tractions field in the cell contour domain. The resulting maximum traction values increased by 11% as a consequence of doubling the mechanical properties of the cell domain.

CONCLUSIONS

With the results generated from computations we demonstrate the application of the algorithm and explain how the mechanical properties of both, the cell and the gel, domains are important for arriving to the correct results when using inverse traction force reconstruction algorithms, however, have only a minor effect on the resulting traction values.

A Finite Element Bendo-Tensegrity Model of Eukaryotic Cell

Mechanical interaction of cell with extracellular environment affects its function. The mechanisms by which mechanical stimuli are sensed and transduced into biochemical responses are still not well understood. Considering this, two finite element (FE) bendo-tensegrity models of a cell in different states are proposed with the aim to characterize cell deformation under different mechanical loading conditions: a suspended cell model elucidating the global response of cell in tensile test simulation and an adherent cell model explicating its local response in atomic force microscopy (AFM) indentation simulation. The force-elongation curve obtained from tensile test simulation lies within the range of experimentally obtained characteristics of smooth muscle cells (SMCs) and illustrates a nonlinear increase in reaction force with cell stretching. The force-indentation curves obtained from indentation simulations lie within the range of experimentally obtained curves of embryonic stem cells (ESCs) and exhibit the influence of indentation site on the overall reaction force of cell. Simulation results have demonstrated that actin filaments (AFs) and microtubules (MTs) play a crucial role in the cell stiffness during stretching, whereas actin cortex (AC) along with actin bundles (ABs) and MTs are essential for the cell rigidity during indentation. The proposed models quantify the mechanical contribution of individual cytoskeletal components to cell mechanics and the deformation of nucleus under different mechanical loading conditions. These results can aid in better understanding of structure-function relationships in living cells.

Heterogeneity in The Mechanical Properties of Integrins Determines Mechanotransduction Dynamics in Bone Osteoblasts

Bone cells are exposed to dynamic mechanical stimulation that is transduced into cellular responses by mechanotransduction mechanisms. The extracellular matrix (ECM) provides a physical link between loading and bone cells, where mechanoreceptors, such as integrins, initiate mechanosensation. Though this relationship is well studied, the dynamic interplay between mechanosensation, mechanotransduction and cellular responses is unclear. A hybrid-multiscale model combining molecular, cellular and tissue interactions was developed to examine links between integrins’ mechanosensation and effects on mechanotransduction, ECM modulation and cell-ECM interaction. The model shows that altering integrin mechanosensitivity threshold (MT) increases mechanotransduction durations from hours to beyond 4 days, where bone formation starts. This is relevant to bone, where it is known that a brief stimulating period provides persistent influences for over 24 hours. Furthermore, the model forecasts that integrin heterogeneity, with respect to MT, would be able to induce sustained increase in pERK baseline > 15% beyond 4 days. This is analogous to the emergence of molecular mechanical memory signalling dynamics. Therefore, the model can provide a greater understanding of mechanical adaptation to differential mechanical responses at different times. Given reduction of bone sensitivity to mechanical stimulation with age, these findings may lead towards useful therapeutic targets for upregulation of bone mass.

The consequence of substrates of large-scale rigidity on actin network tension in adherent cells

There is compelling evidence that substrate stiffness affects cell adhesion as well as cytoskeleton organization and contractile activity. This work was designed to study the cytoskeletal contractile activity of single cells plated on micropost substrates of different stiffness using a numerical model simulating the intracellular tension of individual cells. We allowed cells to adhere onto micropost substrates of various rigidities and used experimental traction force data to infer cell contractility using a numerical model. The model shows that higher substrate stiffness leads to an increase in intracellular tension. The strength of this model is its ability to calculate the mechanical state of each cell in accordance to its individual cytoskeletal structure. This is achieved by regenerating a numerical cytoskeleton based on microscope images of the actin network of each cell. The resulting numerical structure consequently represents pulling characteristics on its environment similar to those generated by the cell in-vivo. From actin imaging we can calculate and better understand how forces are transmitted throughout the cell.

Biomechanical Heterogeneity of Living Cells: Comparison between Atomic Force Microscopy and Finite Element Simulation

Atomic force microscopy (AFM) indentation is a popular method for characterizing the micromechanical properties of soft materials such as living cells. However, the mechanical data obtained from deep indentation measurements can be difficult and problematic to interpret as a result of the complex geometry of a cell, the nonlinearity of indentation contact, and constitutive relations of heterogeneous hyperelastic soft components. Living MDA-MB-231 cells were indented by spherical probes to obtain morphological and mechanical data that were adopted to build an accurate finite element model (FEM) for a parametric study. Initially, a 2D-axisymmetric numerical model was constructed with the main purpose of understanding the effect of geometrical and mechanical properties of constitutive parts such as the cell body, nucleus, and lamellipodium. A series of FEM deformation fields were directly compared with atomic force spectroscopy in order to resolve the mechanical convolution of heterogeneous parts and quantify Young's modulus and the geometry of nuclei. Furthermore, a 3D finite element model was constructed to investigate indentation events located far from the axisymmetric geometry. In this framework, the joint FEM/AFM approach has provided a useful methodology and a comprehensive characterization of the heterogeneous structure of living cells, emphasizing the deconvolution of geometrical structure and the true elastic modulus of the cell nucleus.

A Nondimensional Model Reveals Alterations in Nuclear Mechanics upon Hepatitis C Virus Replication

Morphology of the nucleus is an important regulator of gene expression. Nuclear morphology is in turn a function of the forces acting on it and the mechanical properties of the nuclear envelope. Here, we present a two-parameter, nondimensional mechanical model of the nucleus that reveals a relationship among nuclear shape parameters, such as projected area, surface area, and volume. Our model fits the morphology of individual nuclei and predicts the ratio between forces and modulus in each nucleus. We analyzed the changes in nuclear morphology of liver cells due to hepatitis C virus (HCV) infection using this model. The model predicted a decrease in the elastic modulus of the nuclear envelope and an increase in the pre-tension in cortical actin as the causes for the change in nuclear morphology. These predictions were validated biomechanically by showing that liver cells expressing HCV proteins possessed enhanced cellular stiffness and reduced nuclear stiffness. Concomitantly, cells expressing HCV proteins showed downregulation of lamin-A,C and upregulation of β-actin, corroborating the predictions of the model. Our modeling assumptions are broadly applicable to adherent, monolayer cell cultures, making the model amenable to investigate changes in nuclear mechanics due to other stimuli by merely measuring nuclear morphology. Toward this, we present two techniques, graphical and numerical, to use our model for predicting physical changes in the nucleus.

A shear assay study of single normal/breast cancer cell deformation and detachment from poly-di-methyl-siloxane (PDMS) surfaces

This paper presents the results of a combined experimental and analytical/computational study of viscoelastic cell deformation and detachment from poly-di-methyl-siloxane (PDMS) surfaces. Fluid mechanics and fracture mechanics concepts are used to model the detachment of biological cells observed under shear assay conditions. The analytical and computational models are used to compute crack driving forces, which are then related to crack extension during the detachment of normal breast cells and breast cancer cells from PDMS surfaces that are relevant to biomedical implants. The interactions between cells and the extracellular matrix, or the extracellular matrix and the PDMS substrate, are then characterized using force microscopy measurements of the pull-off forces that are used to determine the adhesion energies. Finally, fluorescence microscopy staining of the cytosketelal structures (actin, micro-tubulin and cyto-keratin), transmembrane proteins (vimentin) and the ECM structures (Arginin Glycine Aspartate - RGD) is used to show that the detachment of cells during the shear assay experiments occurs via interfacial cracking between (between the ECM and the cell membranes) with a high incidence of crack bridging by transmembrane vinculin structures that undergo pull-out until they detach from the actin cytoskeletal structure. The implications of the results are discussed for the design of interfaces that are relevant to implantable biomedical devices and normal/cancer tissue.

The Insertion Mechanism of a Living Cell Determined by the Stress Segmentation Effect of the Cell Membrane during the Tip-Cell Interaction

Atomic force microscopy probes are proved to be powerful tools to measure and manipulate the individual cell, providing potential applications for the controlled drug/protein delivery. However, the measured insertion efficiency varies dramatically from 20 to 80%, in some cases, the nanotip can never penetrate the cell membrane no matter how much force is applied to it. Thus, the insertion mechanism of a living cell during the tip-cell interaction must be thoroughly investigated before this technology comes into practical applications. In this work, a multistructural cell model is established to study the tip-membrane interaction. The simulation results show that the stress of the cell membrane can be divided into two stages by the stress segmentation point S. After point S, the stress of the cell membrane increases slightly and most of the indentation force is allocated to the cytoskeleton. This phenomenon is called "stress segmentation effect of the cell membrane," which confirms the hypothesis based on the experimental studies. Moreover, according to the experimental and numerical studies, the hypothesis of the stress segmentation effect also explains the reason that modifying the cell membrane or using the manmade sharpened nanotip can increase the insertion efficiency.

Mitochondrial DNA 3243A>G heteroplasmy is associated with changes in cytoskeletal protein expression and cell mechanics

Mitochondrial and mechanical alterations in cells have both been shown to be hallmarks of human disease. However, little research has endeavoured to establish connections between these two essential features of cells in both functional and dysfunctional situations. In this work, we hypothesized that a specific genetic alteration in mitochondrial function known to cause human disease would trigger changes in cell mechanics. Using a previously characterized set of mitochondrial cybrid cell lines, we examined the relationship between heteroplasmy for the mitochondrial DNA (mtDNA) 3243A>G mutation, the cell cytoskeleton, and resulting cellular mechanical properties. We found that cells with increasing mitochondrial dysfunction markedly differed from one another in gene expression and protein production of various co-regulated cytoskeletal elements. The intracellular positioning and organization of actin also differed across cell lines. To explore the relationship between these changes and cell mechanics, we then measured cellular mechanical properties using atomic force microscopy and found that cell stiffness correlated with gene expression data for known determinants of cell mechanics, γ-actin, α-actinin and filamin A. This work points towards a mechanism linking mitochondrial genetics to single-cell mechanical properties. The transcriptional and structural regulation of cytoskeletal components by mitochondrial function may explain why energetic and mechanical alterations often coexist in clinical conditions.

Morphological and Mechanical Properties of Osteosarcoma Microenvironment Cells Explored by Atomic Force Microscopy

Cell mechanical properties that depend on cytoskeleton architecture are critical to the mechanotransduction process, and have great potential for cancer diagnosis and therapy. In this study, the morphological and mechanical properties of typical osteosarcoma microenvironment cells, including mesenchymal stem cells (MSC), normal human osteoblast cells (NHOst) and osteosarcoma cells (MG-63), were compared using atomic force microscopy (AFM). The MG-63 cells were smaller and thicker than the MSC and NHOst cells. The membrane roughness of MG-63 cells was higher than that of MSC and NHOst cells. The MG-63 cells had lower stiffness than their normal counterparts due to their reduced organization of the cytoskeleton structure. The cell stiffness influenced the mechanotransduction. The MG-63 cells had a lower percentage of nuclear YAP/TAZ compared with the MSC and NHOst cells. The F-actin assembly was disrupted by the cytochalasin D (cyto D) treatment used to investigate its influence on mechanotransduction. Disruption of the cytoskeleton leaded to a decrease of the cell stiffness, and reduced the nuclear YAP/TAZ percentage, indicating its inhibition in the cell mechanotransduction process. This study would shed light on the development of a novel cancer diagnosis strategy and would contribute to reveal the relationship between the cytoskeleton structure and the cell mechanical properties.

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.

Cellular mechanosensitivity to substrate stiffness decreases with increasing dissimilarity to cell stiffness

Computational modelling has received increasing attention to investigate multi-scale coupled problems in micro-heterogeneous biological structures such as cells. In the current study, we investigated for a single cell the effects of (1) different cell-substrate attachment (2) and different substrate modulus [Formula: see text] on intracellular deformations. A fibroblast was geometrically reconstructed from confocal micrographs. Finite element models of the cell on a planar substrate were developed. Intracellular deformations due to substrate stretch of [Formula: see text], were assessed for: (1) cell-substrate attachment implemented as full basal contact (FC) and 124 focal adhesions (FA), respectively, and [Formula: see text]140 KPa and (2) [Formula: see text], 140, 1000, and 10,000 KPa, respectively, and FA attachment. The largest strains in cytosol, nucleus and cell membrane were higher for FC (1.35[Formula: see text], 0.235[Formula: see text] and 0.6[Formula: see text]) than for FA attachment (0.0952[Formula: see text], 0.0472[Formula: see text] and 0.05[Formula: see text]). For increasing [Formula: see text], the largest maximum principal strain was 4.4[Formula: see text], 5[Formula: see text], 5.3[Formula: see text] and 5.3[Formula: see text] in the membrane, 9.5[Formula: see text], 1.1[Formula: see text], 1.2[Formula: see text] and 1.2[Formula: see text] in the cytosol, and 4.5[Formula: see text], 5.3[Formula: see text], 5.7[Formula: see text] and 5.7[Formula: see text] in the nucleus. The results show (1) the importance of representing FA in cell models and (2) higher cellular mechanical sensitivity for substrate stiffness changes in the range of cell stiffness. The latter indicates that matching substrate stiffness to cell stiffness, and moderate variation of the former is very effective for controlled variation of cell deformation. The developed methodology is useful for parametric studies on cellular mechanics to obtain quantitative data of subcellular strains and stresses that cannot easily be measured experimentally.

Cancer cell mechanics with altered cytoskeletal behavior and substrate effects: A 3D finite element modeling study

A robust computational model of a cancer cell is presented using finite element modeling. The model accurately captures nuances of the various components of the cellular substructure. The role of degradation of cytoskeleton on overall elastic properties of the cancer cell is reported. The motivation for degraded cancer cellular substructure, the cytoskeleton is the observation that the innate mechanics of cytoskeleton is disrupted by various anti-cancer drugs as therapeutic treatments for the destruction of the cancer tumors. We report a significant influence on the degradation of the cytoskeleton on the mechanics of cancer cell. Further, a simulations based study is reported where we evaluate mechanical properties of the cancer cell attached to a variety of substrates. The loading of the cancer cell is less influenced by nature of the substrate, but low modulus substrates such as osteoblasts and hydrogels indicate a significant change in unloading behavior and also the plastic deformation. Overall, softer substrates such as osteoblasts and other bone cells result in a much altered unloading response as well as significant plastic deformation. These substrates are relevant to metastasis wherein certain type of cancers such as prostate and breast cancer cells migrate to the bone and colonize through mesenchymal to epithelial transition. The modeling study presented here is an important first step in the development of strong predictive methodologies for cancer progression.

Mechanobiological modelling of tendons: Review and future opportunities

Tendons are adapted to carry large, repeated loads and are clinically important for the maintenance of musculoskeletal health in an increasing, actively ageing population, as well as in elite athletes. Tendons are known to adapt to mechanical loading. Also, their healing and disease processes are highly sensitive to mechanical load. Computational modelling approaches developed to capture this mechanobiological adaptation in tendons and other tissues have successfully addressed many important scientific and clinical issues. The aim of this review is to identify techniques and approaches that could be further developed to address tendon-related problems. Biomechanical models are identified that capture the multi-level aspects of tendon mechanics. Continuum whole tendon models, both phenomenological and microstructurally motivated, are important to estimate forces during locomotion activities. Fibril-level microstructural models are documented that can use these estimated forces to detail local mechanical parameters relevant to cell mechanotransduction. Cell-level models able to predict the response to such parameters are also described. A selection of updatable mechanobiological models is presented. These use mechanical signals, often continuum tissue level, along with rules for tissue change and have been applied successfully in many tissues to predict in vivo and in vitro outcomes. Signals may include scalars derived from the stress or strain tensors, or in poroelasticity also fluid velocity, while adaptation may be represented by changes to elastic modulus, permeability, fibril density or orientation. So far, only simple analytical approaches have been applied to tendon mechanobiology. With the development of sophisticated computational mechanobiological models in parallel with reporting more quantitative data from in vivo or clinical mechanobiological studies, for example, appropriate imaging, biochemical and histological data, this field offers huge potential for future development towards clinical applications.

The effect of equiaxial stretching on the osteogenic differentiation and mechanical properties of human adipose stem cells

Although mechanical cues are known to affect stem cell fate and mechanobiology, the significance of such stimuli on the osteogenic differentiation of human adipose stem cells (hASCs) remains unclear. In this study, we investigated the effect of long-term mechanical stimulation on the attachment, osteogenic differentiation and mechanical properties of hASCs. Tailor-made, pneumatic cell stretching devices were used to expose hASCs to cyclic equiaxial stretching in osteogenic medium. Cell attachment and focal adhesions were visualised using immunocytochemical vinculin staining on days 3 and 6, and the proliferation and alkaline phosphatase activity, as a sign of early osteogenic differentiation, were analysed on days 0, 6 and 10. Furthermore, the mechanical properties of hASCs, in terms of apparent Young's modulus and normalised contractility, were obtained using a combination of atomic force microscopy based indentation and computational approaches. Our results indicated that cyclic equiaxial stretching delayed proliferation and promoted osteogenic differentiation of hASCs. Stretching also reduced cell size and intensified focal adhesions and actin cytoskeleton. Moreover, cell stiffening was observed during osteogenic differentiation and especially under mechanical stimulation. These results suggest that cyclic equiaxial stretching modifies cell morphology, focal adhesion formation and mechanical properties of hASCs. This could be exploited to enhance osteogenic differentiation.

Photothermally triggered actuation of hybrid materials as a new platform for in vitro cell manipulation

Mechanical forces in the cell's natural environment have a crucial impact on growth, differentiation and behaviour. Few areas of biology can be understood without taking into account how both individual cells and cell networks sense and transduce physical stresses. However, the field is currently held back by the limitations of the available methods to apply physiologically relevant stress profiles on cells, particularly with sub-cellular resolution, in controlled in vitro experiments. Here we report a new type of active cell culture material that allows highly localized, directional and reversible deformation of the cell growth substrate, with control at scales ranging from the entire surface to the subcellular, and response times on the order of seconds. These capabilities are not matched by any other method, and this versatile material has the potential to bridge the performance gap between the existing single cell micro-manipulation and 2D cell sheet mechanical stimulation techniques.

Review on experiment-based two- and three-dimensional models for wound healing

Traumatic and chronic wounds are a considerable medical challenge that affects many populations and their treatment is a monetary and time-consuming burden in an ageing society to the medical systems. Because wounds are very common and their treatment is so costly, approaches to reveal the responses of a specific wound type to different medical procedures and treatments could accelerate healing and reduce patient suffering. The effects of treatments can be forecast using mathematical modelling that has the predictive power to quantify the effects of induced changes to the wound-healing process. Wound healing involves a diverse and complex combination of biophysical and biomechanical processes. We review a wide variety of contemporary approaches of mathematical modelling of gap closure and wound-healing-related processes, such as angiogenesis. We provide examples of the understanding and insights that may be garnered using those models, and how those relate to experimental evidence. Mathematical modelling-based simulations can provide an important visualization tool that can be used for illustrational purposes for physicians, patients and researchers.

Single cell active force generation under dynamic loading - Part II: Active modelling insights

In Part I of this two-part study a novel single cell AFM experimental investigation reveals a complex force-strain response of cells to cyclic loading. The biomechanisms underlying such complex behaviour cannot be fully understood without a detailed mechanistic analysis incorporating the key features of active stress generation and remodelling of the actin cytoskeleton. In order to simulate untreated contractile cells an active bio-chemo-mechanical model is developed, incorporating the key features of stress fibre (SF) remodelling and active tension generation. It is demonstrated that a fading memory SF contractility model accurately captures the transient response of cells to dynamic loading. Simulations reveal that high stretching forces during unloading half-cycles (probe retraction) occur due to tension actively generated by axially oriented SFs. On the other hand, hoop oriented SFs generate tension during loading half-cycles, providing a coherent explanation for the elevated compression resistance of contractile cells. Finally, it is also demonstrated that passive non-linear visco-hyperelastic material laws, traditionally used to simulate cell mechanical behaviour, are not appropriate for untreated contractile cells, and their use should be limited to the simulation of cells in which the active force generation machinery of the actin cytoskeleton has been chemically disrupted. In summary, our active modelling framework provides a coherent understanding of the biomechanisms underlying the complex patterns of experimentally observed single cell force generation presented in the first part of this study.

STATEMENT OF SIGNIFICANCE

A novel computational investigation into the active and passive response of cells to dynamic loading is performed. An active formulation that considers key features of actin cytoskeleton active contractility and remodelling throughout the cytoplasm is implemented. Simulations provide new insights into the sub-cellular biomechanical response, providing a coherent explanation for the complex patterns of cell force uncovered experimentally in the first part of this study. Our computational models also reveal that passive non-linear visco-hyperelastic material laws, traditionally used to simulate cell mechanical behaviour, are not appropriate for untreated contractile cells, and their use should be limited to the simulation of cells in which the active force generation machinery of the actin cytoskeleton has been chemically disrupted.