A Tensegrity Model of Cell Reorientation on Cyclically Stretched Substrates

Effects of Contact Surface Shape on Dynamic Lifetime and Strength of Molecular Bond Clusters under Displacement- and Force-Controlled Loading Conditions

Many experimental and theoretical studies have shown that the mechanical properties of cells and the extracellular matrix can significantly affect the lifetime and strength of the adhesion clusters of molecular bonds. However, there are few studies on how the shape of the contact surface affects the lifetime and strength of the adhesion clusters of molecular bonds, especially theoretical studies in this area. An idealized model of focal adhesion is adopted, in which two rigid media are bonded together by an array of receptor-ligand bonds modeled as Hookean springs on a complex surface topography, which is described by three parameters: the surface shape factor β, the length of a single identical surface shape L, and the amplitude of surface shapes w. In this study, systematic Monte Carlo simulations of this model are conducted to study the lifetime of the molecular bond cluster under linear incremental force loading and the strength of the molecular bond cluster under linear incremental displacement loading. We find that both small surface shape amplitudes and large surface shape factors will increase the lifetime and strength of the adhesion cluster, whereas the length of a single surface shape causes oscillations in the lifetime and strength of the cluster, and this oscillation amplitude is affected by the surface shape amplitude and the factor. At the same time, we also find that the pretension in the cluster will play a dominant role in the adhesion strength under large amplitudes and small factors of surface shapes. The physical mechanisms behind these phenomena are that the changes of the length of a single surface shape, the amplitude of surface shapes, and the surface shape factor cause the changes of stress concentration in the adhesion region, bond affinity, and the number of similar affinity bonds.

Anisotropic power-law viscoelasticity of living cells is dominated by cytoskeletal network structure

During physiological and pathological processes, cells experience significant morphological alterations with the re-arrangement of cytoskeletal filaments, resulting in anisotropic viscoelasticity. Here, a structure-based cell model is proposed to study the anisotropic viscoelastic mechanical behaviors of living cells. We investigate how cell shape affects its creep responses in longitudinal and perpendicular directions. It is shown that cells exhibit power-law rheological behavior in both longitudinal and perpendicular directions under step stress, with a more solid-like behavior along the longitudinal direction. We reveal that the cell volume and cytoskeletal filament orientation, which have been neglected in most existing models, play a critical role in regulating cellular anisotropic viscoelasticity. The stiffness of the cell in both directions increases linearly with increasing its aspect ratio, due to the decrease of cell volume. Moreover, the increase in the cell's aspect ratio produces the aggregation of cytoskeletal filaments along the longitudinal direction, resulting in higher stiffness in this direction. It is also shown that the increase in cell's aspect ratio corresponds to a process of cellular ordering, which can be quantitatively characterized by the orientational entropy of cytoskeletal filaments. In addition, we present a simple yet robust method to establish the relationship between cell's aspect ratio and cell volume, thus providing a theoretical framework to capture the anisotropic viscoelastic behavior of cells. This study suggests that the structure-based cell models may be further developed to investigate cellular rheological responses to external mechanical stimuli and may be extended to the tissue scale. STATEMENT OF SIGNIFICANCE: The viscoelastic behaviors of cells hold significant importance in comprehending the roles of mechanical forces in embryo development, invasion, and metastasis of cancer cells. Here, a structure-based cell model is proposed to study the anisotropic viscoelastic mechanical behaviors of living cells. Our study highlights the crucial role of previously neglected factors, such as cell volume and cytoskeletal filament orientation, in regulating cellular anisotropic viscoelasticity. We further propose an orientational entropy of cytoskeletal filaments to quantitatively characterize the ordering process of cells with increasing aspect ratios. Moreover, we derived the analytical interrelationships between cell aspect ratio, cell stiffness, cell volume, and cytoskeletal fiber orientation. This study provides a theoretical framework to describe the anisotropic viscoelastic mechanical behavior of cells.

Mechanical guidance to self-organization and pattern formation of stem cells

The self-organization of stem cells (SCs) constitutes the fundamental basis of the development of biological organs and structures. SC-driven patterns are essential for tissue engineering, yet unguided SCs tend to form chaotic patterns, impeding progress in biomedical engineering. Here, we show that simple geometric constraints can be used as an effective mechanical modulation approach that promotes the development of controlled self-organization and pattern formation of SCs. Using the applied SC guidance with geometric constraints, we experimentally uncover a remarkable deviation in cell aggregate orientation from a random direction to a specific orientation. Subsequently, we propose a dynamic mechanical framework, including cells, the extracellular matrix (ECM), and the culture environment, to characterize the specific orientation deflection of guided cell aggregates relative to initial geometric constraints, which agrees well with experimental observation. Based on this framework, we further devise various theoretical strategies to realize complex biological patterns, such as radial and concentric structures. Our study highlights the key role of mechanical factors and geometric constraints in governing SCs' self-organization. These findings yield critical insights into the regulation of SC-driven pattern formation and hold great promise for advancements in tissue engineering and bioactive material design for regenerative application.

Bending-activated biotensegrity structure enables female Megarhyssa to cross the barrier of Euler's critical force

The parasitic female Megarhyssa has a hair-like ovipositor capable of withstanding a penetration force 10 times greater than Euler's critical force, using a reciprocating penetration method. Understanding and replicating this penetration mechanism may notably broaden the application scenarios of artificial slender elements. Here, we show that the Megarhyssa's stretched intersegmental membrane and precurved abdomen activate the multipart ovipositor as a biotensegrity structure. The ovipositor's first and second valvulae alternately retract and protract, with each retracted valvula forming a tension network to support the other under compression, resulting in an exponentially increased critical force. We validated this mechanism in a multipart flexible microneedle that withstood a penetration force of 2.5× Euler's critical force and in a lightweight industrial robot that achieved intrinsic safety through its ideal dual-stiffness characteristic. This finding could potentially elucidate the high efficiency of insect probes and inspire more efficient and safer engineering designs.

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

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

Modelling Cell Orientation Under Stretch: The Effect of Substrate Elasticity

When cells are seeded on a cyclically deformed substrate like silicon, they tend to reorient their major axis in two ways: either perpendicular to the main stretching direction, or forming an oblique angle with it. However, when the substrate is very soft such as a collagen gel, the oblique orientation is no longer observed, and the cells align either along the stretching direction, or perpendicularly to it. To explain this switch, we propose a simplified model of the cell, consisting of two elastic elements representing the stress fiber/focal adhesion complexes in the main and transverse directions. These elements are connected by a torsional spring that mimics the effect of crosslinking molecules among the stress fibers, which resist shear forces. Our model, consistent with experimental observations, predicts that there is a switch in the asymptotic behaviour of the orientation of the cell determined by the stiffness of the substratum, related to a change from a supercritical bifurcation scenario, whereby the oblique configuration is stable for a sufficiently large stiffness, to a subcritical bifurcation scenario at a lower stiffness. Furthermore, we investigate the effect of cell elongation and find that the region of the parameter space leading to an oblique orientation decreases as the cell becomes more elongated. This implies that elongated cells, such as fibroblasts and smooth muscle cells, are more likely to maintain an oblique orientation with respect to the main stretching direction. Conversely, rounder cells, such as those of epithelial or endothelial origin, are more likely to switch to a perpendicular or parallel orientation on soft substrates.

A Statistical Mechanics Approach to Describe Cell Reorientation Under Stretch

Experiments show that when a monolayer of cells cultured on an elastic substratum is subject to a cyclic stretch, cells tend to reorient either perpendicularly or at an oblique angle with respect to the main stretching direction. Due to stochastic effects, however, the distribution of angles achieved by the cells is broader and, experimentally, histograms over the interval [Formula: see text] are usually reported. Here we will determine the evolution and the stationary state of probability density functions describing the statistical distribution of the orientations of the cells using Fokker-Planck equations derived from microscopic rules for describing the reorientation process of the cell. As a first attempt, we shall use a stochastic differential equation related to a very general elastic energy that the cell tries to minimize and, we will show that the results of the time integration and of the stationary state of the related forward Fokker-Planck equation compare very well with experimental results obtained by different researchers. Then, in order to model more accurately the microscopic process of cell reorientation and to shed light on the mechanisms performed by cells that are subject to cyclic stretch, we consider discrete in time random processes that allow to recover Fokker-Planck equations through classical tools of kinetic theory. In particular, we shall introduce a model of reorientation as a function of the rotation angle as a result of an optimal control problem. Also in this latter case the results match very well with experiments.

A machine learning-based probabilistic computational framework for uncertainty quantification of actuation of clustered tensegrity structures

Clustered tensegrity structures integrated with continuous cables are lightweight, foldable, and deployable. Thus, they can be used as flexible manipulators or soft robots. The actuation process of such soft structure has high probabilistic sensitivity. It is essential to quantify the uncertainty of actuated responses of the tensegrity structures and to modulate their deformation accurately. In this work, we propose a comprehensive data-driven computational approach to study the uncertainty quantification (UQ) and probability propagation in clustered tensegrity structures, and we have developed a surrogate optimization model to control the flexible structure deformation. An example of clustered tensegrity beam subjected to a clustered actuation is presented to demonstrate the validity of the approach and its potential application. The three main novelties of the data-driven framework are: (1) The proposed model can avoid the difficulty of convergence in nonlinear Finite Element Analysis (FEA), by two machine learning methods, the Gauss Process Regression (GPR) and Neutral Network (NN). (2) A fast real-time prediction on uncertainty propagation can be achieved by the surrogate model, and (3) Optimization of the actuated deformation comes true by using both Sequence Quadratic Programming (SQP) and Bayesian optimization methods. The results have shown that the proposed data-driven computational approach is powerful and can be extended to other UQ models or alternative optimization objectives.

Cellular orientational fluctuations, rotational diffusion and nematic order under periodic driving

The ability of living cells to sense the physical properties of their microenvironment and to respond to dynamic forces acting on them plays a central role in regulating their structure, function and fate. Of particular importance is the cellular sensitivity and response to periodic driving forces in noisy environments, encountered in vital physiological conditions such as heart beating, blood vessel pulsation and breathing. Here, we first test and validate two predictions of a mean-field theory of cellular reorientation under periodic driving, which combines the minimization of cellular anisotropic elastic energy with active remodeling forces. We then extend the mean-field theory to include uncorrelated, additive nonequilibrium fluctuations, and show that the theory quantitatively agrees with the experimentally observed stationary probability distributions of the cell body orientation, under a range of biaxial periodic driving forces. The fluctuations theory allows the extraction of the dimensionless active noise amplitude of various cell types, and consequently their rotational diffusion coefficient. We then focus on intra-cellular nematic order, i.e. on orientational fluctuations of actin stress fibers around the cell body orientation, and show experimentally that intra-cellular nematic order increases with both the magnitude of the driving forces and the biaxiality strain ratio. These results are semi-quantitatively explained by applying the same cell body fluctuations theory to orientationally correlated actin stress fiber domains. Finally, an estimate of the energy scale of cellular orientational fluctuations for one cell type is shown to be about six order of magnitude larger than the thermal energy at room temperature. The implications of our findings, which make the quantitative analysis of cell mechanosensitivity more accessible, are discussed.

The influence of nanotopography on cell behaviour through interactions with the extracellular matrix – A review

Nanotopography presents an effective physical approach for biomaterial cell manipulation mediated through material-extracellular matrix interactions. The extracellular matrix that exists in the cellular microenvironment is crucial for guiding cell behaviours, such as determination of integrin ligation and interaction with growth factors. These interactions with the extracellular matrix regulate downstream mechanotransductive pathways, such as rearrangements in the cytoskeleton and activation of signal cascades. Protein adsorption onto nanotopography strongly influences the conformation and distribution density of extracellular matrix and, therefore, subsequent cell responses. In this review, we first discuss the interactive mechanisms of protein physical adsorption on nanotopography. Secondly, we summarise advances in creating nanotopographical features to instruct desired cell behaviours. Lastly, we focus on the cellular mechanotransductive pathways initiated by nanotopography. This review provides an overview of the current state-of-the-art designs of nanotopography aiming to provide better biomedical materials for the future.

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A comprehensive overview of nanotopography fabrication, and nanotopography regulates various cell behaviours.

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The interactive physical adsorption between nanotopography and extracellular matrix.

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Nanotopography initiates the cellular mechanotransductive pathways and downstream signalling cascades.

Chronic lung diseases: entangled in extracellular matrix

The extracellular matrix (ECM) is the scaffold that provides structure and support to all organs, including the lung; however, it is also much more than this. The ECM provides biochemical and biomechanical cues to cells that reside or transit through this micro-environment, instructing their responses. The ECM structure and composition changes in chronic lung diseases; how such changes impact disease pathogenesis is not as well understood. Cells bind to the ECM through surface receptors, of which the integrin family is one of the most widely recognised. The signals that cells receive from the ECM regulate their attachment, proliferation, differentiation, inflammatory secretory profile and survival. There is extensive evidence documenting changes in the composition and amount of ECM in diseased lung tissues. However, changes in the topographical arrangement, organisation of the structural fibres and stiffness (or viscoelasticity) of the matrix in which cells are embedded have an undervalued but strong impact on cell phenotype. The ECM in diseased lungs also changes in physical and biomechanical ways that drive cellular responses. The characteristics of these environments alter cell behaviour and potentially orchestrate perpetuation of lung diseases. Future therapies should target ECM remodelling as much as the underlying culprit cells.

Modeling and simulation of cell migration on the basis of force equilibrium

To study cell behavior, we developed a cell model to simulate cell movements and the interacting forces among cells and between cells and obstacles. The developed model simulates several cells simultaneously and examines correlations among characteristic parameters between cells and substrates during migration. We modified Odde's model to develop fundamental model, applied Gillespie's stochastic algorithm to design time during in the migration simulation, and employed Keren's membrane theory to analyze the equilibrium at the leading edges. Thus, the proposed model can analyze stresses due to substrate, the intracellular body, and the external interaction between cells and obstacles. Simulation results indicate that cell-cell interaction depends on the equilibrium between the forces at the leading edge of the membrane, namely the cell-substrate interaction force, cell-cell interaction forces, and the cell membrane force. These results also indicate that the migration direction is dependent on the resultant forces. The membrane force and substrate force directions are "low correlation," and the polymerization rate exhibits "little correlative" with the migration direction. We propose a modified cell migration model for simulating allocation and interaction among multiple cells. This model helps indicate the weightings of characteristic parameters that affect the cell migration direction and velocity.

The effect of fluid shear stress on fibroblasts and stem cells on plane and groove topographies

In this study, we aimed to study the effect of fluid shear stress on fibroblasts and BMSCs on plane and groove topographies. The results showed that 0.6-Hz stress had the greatest influence on the alignment, polarity, migration and adhesion of fibroblasts on plane by increasing the expression of reoriented actin and vinculin; whereas 1.0-Hz stress promoted differentiation of fibroblasts into myofibroblasts by increasing Col-I and α-SMA expression. Interestingly, under the given frequency stress, the groove structure strengthened the above characteristics of fibroblasts beyond adhesion, and promoted differentiation of BMSCs into myofibroblasts. The above results indicate that 0.6 Hz may improve the implant-tissue sealing, while 1.0-Hz stress probably causes the disordered fiber deposition around implants.

Cell orientation under stretch: Stability of a linear viscoelastic model

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

Critical Frequency and Critical Stretching Rate for Reorientation of Cells on a Cyclically Stretched Polymer in a Microfluidic Chip

The ability of cells to sense and respond to mechanical signals from their surrounding microenvironments is one of the key issues in tissue engineering and regeneration, yet a fundamental study of cells with both cell observation and mechanical stimulus is challenging and should be based upon an appropriate microdevice. Herein we designed and fabricated a two-layer microfluidic chip to enable simultaneous observation of live cells and cyclic stretching of an elastic polymer, polydimethylsiloxane (PDMS), with a modified surface for enhanced cell adhesion. Human mesenchymal stem cells (hMSCs) were examined with a series of frequencies from 0.00003 to 2 Hz and varied amplitudes of 2%, 5%, or 10%. The cells with an initial random orientation were confirmed to be reoriented perpendicular to the stretching direction at frequencies greater than a threshold value, which we term critical frequency (f_c); additionally, the critical frequency f_c was amplitude-dependent. We further introduced the concept of critical stretching rate (R_c) and found that this quantity can unify both frequency and amplitude dependences. The reciprocal value of R_c in this study reads 8.3 min, which is consistent with the turnover time of actin filaments reported in the literature, suggesting that the supramolecular relaxation in the cytoskeleton within a cell might be responsible for the underlying cell mechanotransduction. The theoretical calculation of cell reorientation based on a two-dimensional tensegrity model under uniaxial cyclic stretching is well consistent with our experiments. The above findings provide new insight into the crucial role of critical frequency and critical stretching rate in regulating cells on biomaterials under biomechanical stimuli.

Dynamically Re-Organized Collagen Fiber Bundles Transmit Mechanical Signals and Induce Strongly Correlated Cell Migration and Self-Organization

Correlated cell migration in fibrous extracellular matrix (ECM) is important in many biological processes. During migration, cells can remodel the ECM, leading to the formation of mesoscale structures such as fiber bundles. However, how such mesoscale structures regulate correlated single-cells migration remains to be elucidated. Here, using a quasi-3D in vitro model, we investigate how collagen fiber bundles are dynamically re-organized and guide cell migration. By combining laser ablation technique with 3D tracking and active-particle simulations, we definitively show that only the re-organized fiber bundles that carry significant tensile forces can guide strongly correlated cell migration, providing for the first time a direct experimental evidence supporting that matrix-transmitted long-range forces can regulate cell migration and self-organization. This force regulation mechanism can provide new insights for studies on cellular dynamics, fabrication or selection of biomedical materials in tissue repairing, and many other biomedical applications.

A Fully Integrated Arduino-Based System for the Application of Stretching Stimuli to Living Cells and Their Time-Lapse Observation: A Do-It-Yourself Biology Approach

Mechanobiology has nowadays acquired the status of a topic of fundamental importance in a degree in Biological Sciences. It is inherently a multidisciplinary topic where biology, physics and engineering competences are required. A course in mechanobiology should include lab experiences where students can appreciate how mechanical stimuli from outside affect living cell behaviour. Here we describe all the steps to build a cell stretcher inside an on-stage cell incubator. This device allows exposing living cells to a periodic mechanical stimulus similar to what happens in physiological conditions such as, for example, in the vascular system or in the lungs. The reaction of the cells to the periodic mechanical stretching represents a prototype of a mechanobiological signal integrated by living cells. We also provide the theoretical and experimental aspects related to the calibration of the stretcher apparatus at a level accessible to researchers not used to dealing with topics like continuum mechanics and analysis of deformations. We tested our device by stretching cells of two different lines, U87-MG and Balb-3T3 cells, and we analysed and discussed the effect of the periodic stimulus on both cell reorientation and migration. We also discuss the basic aspects related to the quantitative analysis of the reorientation process and of cell migration. We think that the device we propose can be easily reproduced at low-cost within a project-oriented course in the fields of biology, biotechnology and medical engineering.

Proximity-induced electrodeformation and membrane capacitance coupling between cells

Membrane capacitance and transmembrane potential are sensitive to the proximity of neighboring biological cells which eventually induces anisotropic perturbation of the local electric field distribution in a cell assembly and/or a tissue. The development of robust and reliable multiphysics approaches is essential to solve the challenge of analyzing proximity-induced capacitance coupling (CC) between cells. In this study, we ask to what extent this CC is a minor perturbation on the individual cells or whether it can fundamentally affect bio-electromechanical cues. A key component of our continuum electromechanical analysis is the consideration of elastic models of cells under steady state electric field excitation to characterize electrodeformation (ED). Analyzing the difference between the ED force for a pair of cells and its counterpart for a single reference cell allows us to determine a separation distance-orientation angle diagram providing evidence of a separation distance beyond which the electrostatic interactions between a pair of biological cells become inconsequential for the ED. An attenuation-amplification transition of ED force in this diagram suggests that anisotropy induced by the orientation angle of the cell pair relative to the applied electric field direction has a significant influence on ED and CC. We furthermore observe that the shape of this diagram changes when extracellular conductivity is varied. The results obtained are then contrasted with the corresponding diagrams of similar cell configurations under an oscillating electric field excitation below and above the α-dispersion frequency. This investigation may provide new opportunities for further assessment of electromechanical properties of engineered tissues.

Heterogeneous role of integrins in fibroblast response to small cyclic mechanical stimulus generated by a nanoporous gold actuator

It is important to understand the effects of mechanical stimulation on cell behaviors for homeostasis. Many studies have been performed on cell responses to mechanical stimuli, but the mechanosensing mechanism is still under debate. In the present study, experiments employing molecular dynamics (MD) simulations concerning the effects of cyclic mechanical stimulus on cell proliferation were performed based on the hypothesis that mechanosensing depends on integrin types. We used a nanoporous gold (NPG) actuator to prevent transfer of a mechanical stimulus via molecules other than integrins. Surprisingly, a small cyclic strain of only 0.5% enhanced the proliferation of fibroblasts. α₅β₁ and α_vβ₃ integrins showed high sensitivity to the mechanical stimulus, whereas α₁β₁ and α₂β₁ integrins exhibited low mechanosensitivity. The MD simulations showed that different conformational changes of the integrin headpiece induced by binding to the ECM led to a difference in mechanosensitivity between αI and αI-less integrin types. Thus, the present study provides evidence to support the hypothesis and suggests the mechanism for the heterogeneous roles of integrins in mechanosensing.

A nonlinear elastic description of cell preferential orientations over a stretched substrate

The active response of cells to mechanical cues due to their interaction with the environment has been of increasing interest, since it is involved in many physiological phenomena, pathologies, and in tissue engineering. In particular, several experiments have shown that, if a substrate with overlying cells is cyclically stretched, they will reorient to reach a well-defined angle between their major axis and the main stretching direction. Recent experimental findings, also supported by a linear elastic model, indicated that the minimization of an elastic energy might drive this reorientation process. Motivated by the fact that a similar behaviour is observed even for high strains, in this paper we address the problem in the framework of finite elasticity, in order to study the presence of nonlinear effects. We find that, for a very large class of constitutive orthotropic models and with very general assumptions, there is a single linear relationship between a parameter describing the biaxial deformation and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cos ^2\theta _{\mathrm{eq}}$$\end{document}cos2θeq, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta _{\mathrm{eq}}$$\end{document}θeq is the orientation angle of the cell, with the slope of the line depending on a specific combination of four parameters that characterize the nonlinear constitutive equation. We also study the effect of introducing a further dependence of the energy on the anisotropic invariants related to the square of the Cauchy–Green strain tensor. This leads to departures from the linear relationship mentioned above, that are again critically compared with experimental data.

Effect of Cyclic Stretch on Neuron Reorientation and Axon Outgrowth

The directional alignment and outgrowth of neurons is a critical step of nerve regeneration and functional recovery of nerve systems, where neurons are exposed to a complex mechanical environment with subcellular structures such as stress fibers and focal adhesions acting as the key mechanical transducer. In this paper, we investigate the effects of cyclic stretch on neuron reorientation and axon outgrowth with a feasible stretching device that controls stretching amplitude and frequency. Statistical results indicate an evident frequency and amplitude dependence of neuron reorientation, that is, neurons tend to align away from stretch direction when stretching amplitude and frequency are large enough. On the other hand, axon elongation under cyclic stretch is very close to the reference case where neurons are not stretched. A mechanochemical framework is proposed by connecting the evolution of cellular configuration to the microscopic dynamics of subcellular structures, including stress fiber, focal adhesion, and microtubule, yielding theoretical predictions that are consistent with the experimental observations. The theoretical work provides an explanation of the neuron's mechanical response to cyclic stretch, suggesting that the contraction force generated by stress fiber plays an essential role in both neuron reorientation and axon elongation. This combined experimental and theoretical study on stretch-induced neuron reorientation may have potential applications in neurodevelopment and neuron regeneration.

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

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

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.

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.

Finite Element Analysis of the Nanomechanics of Hard Coatings on a Soft Polymer Substrate by a Spherical Indenter

Composite has been widely used in various fields due to its advanced performance. To reveal the relation between the mechanical properties of the composite and that of each individual component, finite element analysis (FEA) has usually been adopted. In this study, in order to predict the mechanical properties of hard coating on a soft polymer, the response of this coating system during nanoindentation was modelled. Various models, such as a viscoelastic model and fitting model, were adopted to analyse the indentation response of this coating system. By varying the substrate properties (i.e., Young’s modulus, viscoelasticity, and Poisson’s ratio), Young’s modulus, energy loss, and the viscoelastic model of the coating system were analysed, and how the mechanical properties of the substrate will affect the indentation response of the coating system was discussed.

Elliptical adhesive contact under biaxial stretching

Adhesive contact of the Hertzian indenter with an incompressible elastic substrate bi-directionally stretched along the indenter principal planes of curvature is considered in the Johnson-Kendall-Roberts theoretical framework. An approximate model is constructed by examining energy release rate conditions only on the edges of the minor and major axes of the contact ellipse. The effect of weak coupling between fracture modes I and II is introduced using a phenomenological mode-mixity function. This study was motivated by the need to model a passive-adhesive mechanism in cell mechanics on stretchable substrates.

Mechanical Model for Catch-Bond-Mediated Cell Adhesion in Shear Flow

Catch bond, whose lifetime increases with applied tensile force, can often mediate rolling adhesion of cells in a hydrodynamic environment. However, the mechanical mechanism governing the kinetics of rolling adhesion of cells through catch-bond under shear flow is not yet clear. In this study, a mechanical model is proposed for catch-bond-mediated cell adhesion in shear flow. The stochastic reaction of bond formation and dissociation is described as a Markovian process, whereas the dynamic motion of cells follows classical analytical mechanics. The steady state of cells significantly depends on the shear rate of flow. The upper and lower critical shear rates required for cell detachment and attachment are extracted, respectively. When the shear rate increases from the lower threshold to the upper threshold, cell rolling became slower and more regular, implying the flow-enhanced adhesion phenomenon. Our results suggest that this flow-enhanced stability of rolling adhesion is attributed to the competition between stochastic reactions of bonds and dynamics of cell rolling, instead of force lengthening the lifetime of catch bonds, thereby challenging the current view in understanding the mechanism behind this flow-enhanced adhesion phenomenon. Moreover, the loading history of flow defining bistability of cell adhesion in shear flow is predicted. These theoretical predictions are verified by Monte Carlo simulations and are related to the experimental observations reported in literature.

Stretching of fibroblast cells on micropatterned gelatin on silicone elastomer

Micropatterned gelatin was formed on the silicone elastomer surface. The micropattern enabled cell alignment, regulation of the cell shape, and endowed the cells with resistance against mechanical stress.

Discovery and design of soft polymeric bio-inspired materials with multiscale simulations and artificial intelligence

Establishing the “Materials 4.0” paradigm requires intimate knowledge of the virtual space in materials design.

Effects of cyclic fluid stress at different frequencies on behaviors of cells incubated on titanium alloy

The orthopedic external fixation is always in dynamic mechanical environment with the somatic movement. We used a self-designed mini oscillator to simulate this condition by providing the reciprocating cyclic fluid stress, and observed the behavioral responses of fibroblasts implanted on titanium alloy plane to the stress at different frequencies, including 0.2 Hz, 0.6 Hz, and 1.0 Hz. We found that the cell angle, shape index and expression of vinculin were mostly biphasic-dependent with the increase of frequency, with peaks at 0.6 Hz. Whereas the cell area, expression of Col-I and α-SMA were mainly affected by the 1.0 Hz stress. Interestingly, 1.0 Hz stress also promoted Col-I expression of bone marrow mesenchymal stem cells (BMSCs), although it did not increase α-SMA. These results reveal that 0.6 Hz stress improves the alignment, polarity and adherence of fibroblasts on titanium alloy substrates, thus improving the sealing of implants; the 1.0 Hz force activates the differentiation of fibroblasts into myofibroblasts and increases collagen produced by stem cells, which probably cause the formation of fibrous capsules around implants.

Enumeration-screening method for the design of simple polygonal tensegrities

Tensegrities, consisting of axially pre-compressed bars and pre-stretched strings, hold broad applications in the design of, for instance, architectures, soft robotics and metamaterials. In this paper, we propose an enumeration-screening method to design planar tensegrities of simple polygonal shapes. In such a polygonal tensegrity, the strings are joined pair-wise to form a simple polygon (a planar shape consisting of straight, non-intersecting line segments) and only one bar is added at each node. The total number of simple polygonal tensegrities designed by this scheme increases exponentially with the number of bars. Moreover, we demonstrate that each of these designed topologies can produce a self-equilibrated and stable tensegrity configuration. This work helps understand the topological features of simple polygonal tensegrities, which can be used as elementary cells to design some novel two- and three-dimensional tensegrity structures.

Mechanical Roles of F-Actin in the Differentiation of Stem Cells: A Review

In the development and differentiation of stem cells, mechanical forces associated with filamentous actin (F-actin) play a crucial role. The present review aims to reveal the relationship among the chemical components, microscopic structures, mechanical properties, and biological functions of F-actin. Particular attention is given to the functions of the cytoplasmic and nuclear microfilament cytoskeleton and their regulation mechanisms in the differentiation of stem cells. The distributions of different types of actin monomers in mammal cells and the functions of actin-binding proteins are summarized. We discuss how the fate of stem cells is regulated by intra/extracellular mechanical and chemical cues associated with microfilament-related proteins, intercellular adhesion molecules, etc. In addition, we also address the differentiation-induced variation in the stiffness of stem cells and the correlation between the fate and geometric shape change of stem cells. This review not only deepens our understanding of the biophysical mechanisms underlying the fates of stem cells under different culture conditions but also provides inspirations for the tissue engineering of stem cells.

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

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

Universal Kinetics of the Onset of Cell Spreading on Substrates of Different Stiffness

When plated onto substrates, cell morphology and even stem-cell differentiation are influenced by the stiffness of their environment. Stiffer substrates give strongly spread (eventually polarized) cells with strong focal adhesions and stress fibers; very soft substrates give a less developed cytoskeleton and much lower cell spreading. The kinetics of this process of cell spreading is studied extensively, and important universal relationships are established on how the cell area grows with time. Here, we study the population dynamics of spreading cells, investigating the characteristic processes involved in the cell response to the substrate. We show that unlike the individual cell morphology, this population dynamics does not depend on the substrate stiffness. Instead, a strong activation temperature dependence is observed. Different cell lines on different substrates all have long-time statistics controlled by the thermal activation over a single energy barrier ΔG ≈ 18 kcal/mol, whereas the early-time kinetics follows a power law ∼t⁵. This implies that the rate of spreading depends on an internal process of adhesion complex assembly and activation; the operational complex must have five component proteins, and the last process in the sequence (which we believe is the activation of focal adhesion kinase) is controlled by the binding energy ΔG.

Mechanical characterization of disordered and anisotropic cellular monolayers

We consider a cellular monolayer, described using a vertex-based model, for which cells form a spatially disordered array of convex polygons that tile the plane. Equilibrium cell configurations are assumed to minimize a global energy defined in terms of cell areas and perimeters; energy is dissipated via dynamic area and length changes, as well as cell neighbor exchanges. The model captures our observations of an epithelium from a Xenopus embryo showing that uniaxial stretching induces spatial ordering, with cells under net tension (compression) tending to align with (against) the direction of stretch, but with the stress remaining heterogeneous at the single-cell level. We use the vertex model to derive the linearized relation between tissue-level stress, strain, and strain rate about a deformed base state, which can be used to characterize the tissue’s anisotropic mechanical properties; expressions for viscoelastic tissue moduli are given as direct sums over cells. When the base state is isotropic, the model predicts that tissue properties can be tuned to a regime with high elastic shear resistance but low resistance to area changes, or vice versa.

Capillary network formation from dispersed endothelial cells: Influence of cell traction, cell adhesion, and extracellular matrix rigidity

The formation of a functional vascular network depends on biological, chemical, and physical processes being extremely well coordinated. Among them, the mechanical properties of the extracellular matrix and cell adhesion are fundamental to achieve a functional network of endothelial cells, able to fully cover a required domain. By the use of a Cellular Potts Model and Finite Element Method it is shown that there exists a range of values of endothelial traction forces, cell-cell adhesion, and matrix rigidities where the network can spontaneously be formed, and its properties are characterized. We obtain the analytical relation that the minimum traction force required for cell network formation must obey. This minimum value for the traction force is approximately independent on the considered cell number and cell-cell adhesion. We quantify how these two parameters influence the morphology of the resulting networks (size and number of meshes).

Control of cellular adhesion and myofibroblastic character with sub-micrometer magnetoelastic vibrations

The effect of sub-cellular mechanical loads on the behavior of fibroblasts was investigated using magnetoelastic (ME) materials, a type of material that produces mechanical vibrations when exposed to an external magnetic AC field. The integration of this functionality into implant surfaces could mitigate excessive fibrotic responses to many biomedical devices. By changing the profiles of the AC magnetic field, the amplitude, duration, and period of the applied vibrations was altered to understand the effect of each parameter on cell behavior. Results indicate fibroblast adhesion depends on the magnitude and total number of applied vibrations, and reductions in proliferative activity, cell spreading, and the expression of myofibroblastic markers occur in response to the vibrations induced by the ME materials. These findings suggest that the subcellular amplitude mechanical loads produced by ME materials could potentially remotely modulate myofibroblastic activity and limit undesirable fibrotic development.

Orientations of Cells on Compliant Substrates under Biaxial Stretches: A Theoretical Study

Mechanical cues from the microenvironments play a regulating role in many physiological and pathological processes, such as stem cell differentiation and cancer cell metastasis. Experiments showed that cells adhered on a compliant substrate may change orientation with an externally applied strain in the substrate. By accounting for actin polymerization, actin retrograde flow, and integrin binding dynamics, here we develop a mechanism-based tensegrity model to study the orientations of polarized cells on a compliant substrate under biaxial stretches. We show that the cell can actively regulate its mechanical state by generating different traction force levels along its polarized direction. Under static or ultralow-frequency cyclic stretches, stretching a softer substrate leads to a higher increase in the traction force and induces a narrower distribution of cell alignment. Compared to static loadings, high-frequency cyclic loadings have a more significant influence on cell reorientation on a stiff substrate. In addition, the width of the cellular angular distribution scales inversely with the stretch amplitude under both static and cyclic stretches. Our results are in agreement with a wide range of experimental observations, and provide fundamental insights into the functioning of cellular mechanosensing systems.

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.

Elastic Properties of Pore-Spanning Apical Cell Membranes Derived from MDCK II Cells

The mechanical response of adherent, polarized cells to indentation is frequently attributed to the presence of an endogenous actin cortex attached to the inner leaflet of the plasma membrane. Here, we scrutinized the elastic properties of apical membranes separated from living cells and attached to a porous mesh in the absence of intracellular factors originating from the cytosol, organelles, the substrate, neighbors, and the nucleus. We found that a tension-based model describes the data very well providing essentially the prestress of the shell generated by adhesion of the apical membrane patches to the pore rim and the apparent area compressibility modulus, an intrinsic elastic modulus modulated by the surface excess stored in membrane reservoirs. Removal of membrane-associated proteins by proteases decreases the area compressibility modulus, whereas fixation and cross-linking of proteins with glutaraldehyde increases it.

Thae use of microfluic spinning fiber as an ophthalmology suture showing the good anastomotic strength control

Adjusting the mechanical strength of a biomaterial to suit its intended application is very important for realizing beneficial outcomes. Microfluidic spinning fiber have been attracting attention recently due to their various advantages, but their mechanical strength has unfortunately not been a subject of concentrated research, and this lack of research has severely limited their applications. In the current work, we showed the mechanical properties of microfibers can be tuned easily and provided a mathematical explanation for how the microfluidic spinning method intrinsically controls the mechanical properties of a microfluidic spinning fiber. But we were also able to adjust the mechanical properties of such fibers in various other ways, including by using biomolecules to coat the fiber or mixing the biomolecules with the primary component of the fiber and by using a customized twisting machine to change the number of single microfiber strands forming the fiber. We used the bundle fiber as an ophthalmology suture that resulted in a porcine eye with a smoother post-operative surface than did a nylon suture. The results showed the possibility that the proposed method can solve current problems of the microfibers in practical applications, and can thus extend the range of applications of these microfibers.

A Dynamic Biochemomechanical Model of Geometry-Confined Cell Spreading

Cell spreading is involved in many physiological and pathological processes. The spreading behavior of a cell significantly depends on its microenvironment, but the biochemomechanical mechanisms of geometry-confined cell spreading remain unclear. A dynamic model is here established to investigate the spreading of cells confined in a finite region with different geometries, e.g., rectangle, ellipse, triangle, and L-shape. This model incorporates both biophysical and biochemical mechanisms, including actin polymerization, integrin-mediated binding, plasma viscoelasticity, and the elasticity of membranes and microtubules. We simulate the dynamic configurational evolution of a cell under different geometric microenvironments, including the angular distribution of microtubule forces and the deformation of the nucleus. The results indicate that the positioning of the cell-division plane is affected by its boundary confinement: a cell divides in a plane perpendicular to its minimal principal axis of inertia of area. In addition, the effects of such physical factors as the adhesive bond density, membrane tension, and microtubule number are examined on the cell spreading dynamics. The theoretical predictions show a good agreement with relevant experimental results. This work sheds light on the geometry-confined spreading dynamics of cells and holds potential applications in regulating cell division and designing cell-based sensors.