A chemo-mechanical free-energy-based approach to model durotaxis and extracellular stiffness-dependent contraction and polarization of cells

Cell-matrix feedback controls stretch-induced cellular memory and fibroblast activation

Mechanical stretch can activate long-lived changes in fibroblasts, increasing their contractility and initiating phenotypic transformations. This activation, critical to wound healing and procedures such as skin grafting, increases with mechanical stimulus for cells cultured in two-dimensional but is highly variable in cells in three-dimensional (3D) tissue. Here, we show that static mechanical stretch of cells in 3D tissues can either increase or decrease fibroblast activation depending upon recursive cell-extracellular matrix (ECM) feedback and demonstrate control of this activation through integrated in vitro and mathematical models. ECM viscoelasticity, signaling dynamics, and cell mechanics combine to yield a predictable, but nonmonotonic, relationship between mechanical stretch and long-term cell activation. Results demonstrate that feedback between cells and ECM determine how cells retain memory of mechanical stretch and have direct implications for improving outcomes in skin grafting procedures.

Critical Cell Spacing Drives Phase Transition in Matrix-Mediated Tissue Condensation

Biological tissues exhibit phase transitions governed by mechanical feedback between cells and their extracellular matrix (ECM). We demonstrate through bio-chemo-mechanical modeling that this emergent behavior arises from competing physical effects: increasing matrix stiffness enhances individual cell activation while simultaneously weakening long-range mechanical communication. This competition establishes a critical cell spacing threshold (80-160 µ m) that precisely matches experimental observations across diverse cell types and collagen densities. Our model reveals that the critical stretch ratio at which fibrous networks transition from compliant to strain-stiffening governs this threshold through the formation of tension bands between neighboring cells. These tension bands create a mechanical percolation network that drives the collective phase transition in tissue behavior. Our model explains how fibrous architecture controls emergent mechanical properties in biological systems and offers insight into both the physics of fiber-reinforced composite materials under active stress, and into potential mechanical interventions for fibrotic disorders.

Modeling the mechanosensitive collective migration of cells on the surface and the interior of morphing soft tissues

Cellular contractility, migration, and extracellular matrix (ECM) mechanics are critical for a wide range of biological processes including embryonic development, wound healing, tissue morphogenesis, and regeneration. Even though the distinct response of cells near the tissue periphery has been previously observed in cell-laden microtissues, including faster kinetics and more prominent cell-ECM interactions, there are currently no models that can fully combine coupled surface and bulk mechanics and kinetics to recapitulate the morphogenic response of these constructs. Mailand et al. (Biophys J 117(5):975-986, 2019) had shown the importance of active elastocapillarity in cell-laden microtissues, but modeling the distinct mechanosensitive migration of cells on the periphery and the interior of highly deforming tissues has not been possible thus far, especially in the presence of active elastocapillary effects. This paper presents a framework for understanding the interplay between cellular contractility, migration, and ECM mechanics in dynamically morphing soft tissues accounting for distinct cellular responses in the bulk and the surface of tissues. The major novelty of this approach is that it enables modeling the distinct migratory and contractile response of cells residing on the tissue surface and the bulk, where concurrently the morphing soft tissues undergo large deformations driven by cell contractility. Additionally, the simulation results capture the changes in shape and cell concentration for wounded and intact microtissues, enabling the interpretation of experimental data. The numerical procedure that accounts for mechanosensitive stress generation, large deformations, diffusive migration in the bulk and a distinct mechanism for diffusive migration on deforming surfaces is inspired from recent work on bulk and surface poroelasticity of hydrogels involving elastocapillary effects, but in this work, a two-field weak form is proposed and is able to alleviate numerical instabilities that were observed in the original method that utilized a three-field mixed finite element formulation.

Microtubule forces drive nuclear damage in LMNA cardiomyopathy

Nuclear homeostasis requires a balance of forces between the cytoskeleton and nucleus. Mutations in the LMNA gene, which encodes the nuclear envelope proteins lamin A/C, disrupt this balance by weakening the nuclear lamina. This results in nuclear damage in contractile tissues and ultimately muscle disease. Intriguingly, disrupting the LINC complex that connects the cytoskeleton to the nucleus has emerged as a promising strategy to ameliorate LMNA-associated cardiomyopathy. Yet how LINC complex disruption protects the cardiomyocyte nucleus remains unclear. To address this, we developed an assay to quantify the coupling of cardiomyocyte contraction to nuclear deformation and interrogated its dependence on the nuclear lamina and LINC complex. We found that, surprisingly, the LINC complex was mostly dispensable for transferring contractile strain to the nucleus, and that increased nuclear strain in lamin A/C-deficient cardiomyocytes was not rescued by LINC complex disruption. Instead, LINC complex disruption eliminated the cage of microtubules encircling the nucleus. Disrupting microtubules was sufficient to prevent nuclear damage and rescue cardiac function induced by lamin A/C deficiency. We computationally simulated the stress fields surrounding cardiomyocyte nuclei and show how microtubule forces generate local vulnerabilities that damage lamin A/C-deficient nuclei. Our work pinpoints localized, microtubule-dependent force transmission through the LINC complex as a pathological driver and therapeutic target for LMNA-cardiomyopathy.

A tensegrity-based mechanochemical model for capturing cell oscillation and reorientation

The cytoskeleton, a dynamic network of structural proteins within cells, is essential for cellular deformation and responds to external mechanical cues. Here, based on the structure of the cytoskeleton, combined with the biochemical reactions of the activator RhoA and the inhibitor F-actin, we develop a novel mechanochemical cytoskeleton model to investigate the mechanical behavior of cells. Interestingly, we find that active stress fibers exhibit diverse dynamical modes at specific inhibitor concentration thresholds. The existence of concentration differences and sustained mechanochemical feedback in activators and inhibitors trigger a global oscillation of isolated cells. In addition, under uniaxial and biaxial stretches, activators and inhibitors preferentially diffuse toward the more significantly deformed cytoskeletal elements, and their dynamic interactions regulate the cell to align with the main stretching direction. Our findings, consistent with many experimental results, provide fundamental insights into cytoskeletal remodeling and cellular mechanosensing mechanisms.

Mechano-metabolism of metastatic breast cancer cells in 2D and 3D microenvironments

Cells regulate their shape and metabolic activity in response to the mechano-chemical properties of their microenvironment. To elucidate the impact of matrix stiffness and ligand density on the bioenergetics of mesenchymal cells, we developed a nonequilibrium, active chemo-mechanical model that accounts for the mechanical energy of the cell and matrix, chemical energy from ATP hydrolysis, interfacial energy, and mechano-sensitive regulation of stress fiber assembly through signaling. By integrating the kinetics and energetics of these processes, we define the cell "metabolic potential" that, when minimized, provides testable predictions of cell contractility, shape, and ATP consumption. Specifically, we show that the morphology of MDA-MB-231 breast cancer cells in 3D collagen changes from spherical to elongated to spherical with increasing matrix stiffness, which is consistent with experimental observations. On 2D hydrogels, our model predicts a hemispherical-to-spindle-to-disc shape transition with increasing gel stiffness. In both cases, we show that these shape transitions emerge from competition between the energy of ATP hydrolysis associated with increased contractility that drives cell elongation and the interfacial energy that favors a rounded shape. Furthermore, our model can predict how increased energy demand in stiffer microenvironments is met by AMPK activation, which is confirmed experimentally in both 2D and 3D microenvironments and found to correlate with the upregulation of mitochondrial potential, glucose uptake, and ATP levels, as well as provide estimates of changes in intracellular adenosine nucleotide concentrations with changing environmental stiffness. Overall, we present a framework for relating adherent cell energy levels and contractility through biochemical regulation of underlying physical processes.

Statement of Significance

Increasing evidence indicates that cellular metabolism is regulated by mechanical cues from the extracellular environment. Forces transmitted from the microenvironment activate mechanotransduction pathways in the cell, which trigger a cascade of biochemical events that impact cytoskeletal tension, cellular morphology and energy budget available to the cell. Using a nonequilibrium free energy-based theory, we can predict the ATP consumption, contractility, and shape of mesenchymal cancer cells, as well as how cells regulate energy levels dependent on the mechanosensitive metabolic regulator AMPK. The insights from our model can be used to understand the mechanosensitive regulation of metabolism during metastasis and tumor progression, during which cells experience dynamic changes in their microenvironment and metabolic state.

Vimentin is a key regulator of cell mechanosensing through opposite actions on actomyosin and microtubule networks

The cytoskeleton is a complex network of interconnected biopolymers consisting of actin filaments, microtubules, and intermediate filaments. These biopolymers work in concert to transmit cell-generated forces to the extracellular matrix required for cell motility, wound healing, and tissue maintenance. While we know cell-generated forces are driven by actomyosin contractility and balanced by microtubule network resistance, the effect of intermediate filaments on cellular forces is unclear. Using a combination of theoretical modeling and experiments, we show that vimentin intermediate filaments tune cell stress by assisting in both actomyosin-based force transmission and reinforcement of microtubule networks under compression. We show that the competition between these two opposing effects of vimentin is regulated by the microenvironment stiffness. These results reconcile seemingly contradictory results in the literature and provide a unified description of vimentin's effects on the transmission of cell contractile forces to the extracellular matrix.

Fibrillar adhesion dynamics govern the timescales of nuclear mechano-response via the vimentin cytoskeleton

The cell nucleus is continuously exposed to external signals, of both chemical and mechanical nature. To ensure proper cellular response, cells need to regulate not only the transmission of these signals, but also their timing and duration. Such timescale regulation is well described for fluctuating chemical signals, but if and how it applies to mechanical signals reaching the nucleus is still unknown. Here we demonstrate that the formation of fibrillar adhesions locks the nucleus in a mechanically deformed conformation, setting the mechanical response timescale to that of fibrillar adhesion remodelling (~1 hour). This process encompasses both mechanical deformation and associated mechanotransduction (such as via YAP), in response to both increased and decreased mechanical stimulation. The underlying mechanism is the anchoring of the vimentin cytoskeleton to fibrillar adhesions and the extracellular matrix through plectin 1f, which maintains nuclear deformation. Our results reveal a mechanism to regulate the timescale of mechanical adaptation, effectively setting a low pass filter to mechanotransduction.

Mathematical modeling of active contraction of the human cardiac myocyte: A review

Background and objective

In this present research paper, a mathematical model has been developed to study myocyte contraction in the human cardiac muscle, using the Land model. Different parts of the human heart with a focus on the composition of the myocyte cells have been explored numerically to enabling us to determine the interaction of various parameters in the heart muscle. The main objective of the work is to direct the study of the Land model, which has been exploited to simulate the contraction of real human myocytes.

Methods

Mathematical models has been developed based on the Hill model and Huxley model. Myocyte contraction for different scenarios, such as in isometric tension and isotonic tension have been studied.

Results

It is found that increase in stretch, the peak active tension increases, in line with well-established length-dependent tension generation. Five parameters are selected: [Ca2+]T50, Tref, TRPN50, β0, and β1, which have been varied in between the range of -50%-100%, to examine the isometric effects of each parameter on the behavior of the tension developed in the intact myocyte cells, with the most sensitive parameter being [Ca2+]T50.

Conclusion

In conclusion, it is found that the Land model provides a good platform for the analysis of the active contraction of the human cardiac myocyte.

Emergent mechanical control of vascular morphogenesis

Vascularization is driven by morphogen signals and mechanical cues that coordinately regulate cellular force generation, migration, and shape change to sculpt the developing vascular network. However, it remains unclear whether developing vasculature actively regulates its own mechanical properties to achieve effective vascularization. We engineered tissue constructs containing endothelial cells and fibroblasts to investigate the mechanics of vascularization. Tissue stiffness increases during vascular morphogenesis resulting from emergent interactions between endothelial cells, fibroblasts, and ECM and correlates with enhanced vascular function. Contractile cellular forces are key to emergent tissue stiffening and synergize with ECM mechanical properties to modulate the mechanics of vascularization. Emergent tissue stiffening and vascular function rely on mechanotransduction signaling within fibroblasts, mediated by YAP1. Mouse embryos lacking YAP1 in fibroblasts exhibit both reduced tissue stiffness and develop lethal vascular defects. Translating our findings through biology-inspired vascular tissue engineering approaches will have substantial implications in regenerative medicine.

Physical forces guide curvature sensing and cell migration mode bifurcating

The ability of cells to sense and adapt to curvy topographical features has been implicated in organ morphogenesis, tissue repair, and tumor metastasis. However, how individual cells or multicellular assemblies sense and differentiate curvatures remains elusive. Here, we reveal a curvature sensing mechanism in which surface tension can selectively activate either actin or integrin flows, leading to bifurcating cell migration modes: focal adhesion formation that enables cell crawling at convex front edges and actin cable assembly that pulls cells forward at concave front edges. The molecular flows and curved front morphogenesis are sustained by coordinated cellular tension generation and transmission. We track the molecular flows and mechanical force transduction pathways by a phase-field model, which predicts that multicellular curvature sensing is more efficient than individual cells, suggesting collective intelligence of cells. The unique ability of cells in curvature sensing and migration mode bifurcating may offer insights into emergent collective patterns and functions of living active systems at different length scales.

Endothelium and Subendothelial Matrix Mechanics Modulate Cancer Cell Transendothelial Migration

Cancer cell extravasation, a key step in the metastatic cascade, involves cancer cell arrest on the endothelium, transendothelial migration (TEM), followed by the invasion into the subendothelial extracellular matrix (ECM) of distant tissues. While cancer research has mostly focused on the biomechanical interactions between tumor cells (TCs) and ECM, particularly at the primary tumor site, very little is known about the mechanical properties of endothelial cells and the subendothelial ECM and how they contribute to the extravasation process. Here, an integrated experimental and theoretical framework is developed to investigate the mechanical crosstalk between TCs, endothelium and subendothelial ECM during in vitro cancer cell extravasation. It is found that cancer cell actin-rich protrusions generate complex push-pull forces to initiate and drive TEM, while transmigration success also relies on the forces generated by the endothelium. Consequently, mechanical properties of the subendothelial ECM and endothelial actomyosin contractility that mediate the endothelial forces also impact the endothelium's resistance to cancer cell transmigration. These results indicate that mechanical features of distant tissues, including force interactions between the endothelium and the subendothelial ECM, are key determinants of metastatic organotropism.

Chemo-mechanical diffusion waves explain collective dynamics of immune cell podosomes

Immune cells, such as macrophages and dendritic cells, can utilize podosomes, mechanosensitive actin-rich protrusions, to generate forces, migrate, and patrol for foreign antigens. Individual podosomes probe their microenvironment through periodic protrusion and retraction cycles (height oscillations), while oscillations of multiple podosomes in a cluster are coordinated in a wave-like fashion. However, the mechanisms governing both the individual oscillations and the collective wave-like dynamics remain unclear. Here, by integrating actin polymerization, myosin contractility, actin diffusion, and mechanosensitive signaling, we develop a chemo-mechanical model for podosome dynamics in clusters. Our model reveals that podosomes show oscillatory growth when actin polymerization-driven protrusion and signaling-associated myosin contraction occur at similar rates, while the diffusion of actin monomers drives wave-like coordination of podosome oscillations. Our theoretical predictions are validated by different pharmacological treatments and the impact of microenvironment stiffness on chemo-mechanical waves. Our proposed framework can shed light on the role of podosomes in immune cell mechanosensing within the context of wound healing and cancer immunotherapy.

DNA tension assays reveal that force-dependent integrin activation regulates neurite outgrowth in primary cortical neurons

Biomechanical inputs are ubiquitously present in biological systems and are known to regulate various cell functions. In particular, neural cell development is sensitive to mechanical regulation, as these cells reside in one of the softest microenvironments in the body. To fully characterize and comprehend how mechanical force modulates early neuronal processes, we prepared substrates functionalized with DNA probes displaying integrin ligands, including cRGD and laminin, to quantify integrin-mediated molecular tension during neurite initiation in primary cortical neurons. Our live-cell imaging analysis reveals that integrin-mediated tension force is highly dynamic and distributed across the cell body, with the overall tension signal gradually increasing during neurite outgrowth. Notably, we detected a consistent level of mechanical force (amplitude = 4.7-12 piconewtons, pN) for cell integrin-ligand interactions. Further quantifications reveal that neurons exhibit faster cell spreading and neurite outgrowth upon interacting with ligands functionalized with 4.7 pN relative to 12 pN probes. These findings indicate that the magnitude of integrin-mediated mechanical feedback regulates neuronal activity during early neuritogenesis. Additionally, we observed that mechanical tension is correlated with calcium signaling, since inhibiting calcium influx substantially reduced mechanical tension. Thus, our findings support that the magnitude of integrin-mediated mechanical feedback regulates neuronal activity during early neuritogenesis and that mechanical force is an essential element complementing well-known biochemical regulatory mechanisms orchestrating the integrin activation machinery and controlled neurite outgrowth in cortical neurons.

Feedback between mechanosensitive signaling and active forces governs endothelial junction integrity

The formation and recovery of gaps in the vascular endothelium governs a wide range of physiological and pathological phenomena, from angiogenesis to tumor cell extravasation. However, the interplay between the mechanical and signaling processes that drive dynamic behavior in vascular endothelial cells is not well understood. In this study, we propose a chemo-mechanical model to investigate the regulation of endothelial junctions as dependent on the feedback between actomyosin contractility, VE-cadherin bond turnover, and actin polymerization, which mediate the forces exerted on the cell-cell interface. Simulations reveal that active cell tension can stabilize cadherin bonds, but excessive RhoA signaling can drive bond dissociation and junction failure. While actin polymerization aids gap closure, high levels of Rac1 can induce junction weakening. Combining the modeling framework with experiments, our model predicts the influence of pharmacological treatments on the junction state and identifies that a critical balance between RhoA and Rac1 expression is required to maintain junction stability. Our proposed framework can help guide the development of therapeutics that target the Rho family of GTPases and downstream active mechanical processes.

Unified multiscale theory of cellular mechanical adaptations to substrate stiffness

Rigidity of the extracellular matrix markedly regulates many cellular processes. However, how cells detect and respond to matrix rigidity remains incompletely understood. Here, we propose a unified two-dimensional multiscale framework accounting for the chemomechanical feedback to explore the interrelated cellular mechanosensing, polarization, and migration, which constitute the dynamic cascade in cellular response to matrix stiffness but are often modeled separately in previous theories. By combining integrin dynamics and intracellular force transduction, we show that substrate stiffness can act as a switch to activate or deactivate cell polarization. Our theory quantitatively reproduces rich stiffness-dependent cellular dynamics, including spreading, polarity selection, migration pattern, durotaxis, and even negative durotaxis, reported in a wide spectrum of cell types, and reconciles some inconsistent experimental observations. We find that a specific bipolarized mode can determine the optimal substrate stiffness, which enables the fastest cell migration rather than the largest traction forces that cells apply on the substrate. We identify that such a mechanical adaptation stems from the force balance across the whole cell. These findings could yield universal insights into various stiffness-mediated cellular processes within the context of tissue morphogenesis, wound healing, and cancer invasion.

Thermodynamically-motivated chemo-mechanical models and multicellular simulation to provide new insight into active cell and tumour remodelling

Computational models can shape our understanding of cell and tissue remodelling, from cell spreading, to active force generation, adhesion, and growth. In this mini-review, we discuss recent progress in modelling of chemo-mechanical cell behaviour and the evolution of multicellular systems. In particular, we highlight recent advances in (i) free-energy based single cell models that can provide new fundamental insight into cell spreading, cancer cell invasion, stem cell differentiation, and remodelling in disease, and (ii) mechanical agent-based models to simulate large numbers of discrete interacting cells in proliferative tumours. We describe how new biological understanding has emerged from such theoretical models, and the trade-offs and constraints associated with current approaches. Ultimately, we aim to make a case for why theory should be integrated with an experimental workflow to optimise new in-vitro studies, to predict feedback between cells and their microenvironment, and to deepen understanding of active cell behaviour.

The Elephant in the Cell: Nuclear Mechanics and Mechanobiology

The 2021 Summer Biomechanics, Bioengineering, and Biotransport Conference (SB3C) featured a workshop titled "The Elephant in the Room: Nuclear Mechanics and Mechanobiology." The goal of this workshop was to provide a perspective from experts in the field on the current understanding of nuclear mechanics and its role in mechanobiology. This paper reviews the major themes and questions discussed during the workshop, including historical context on the initial methods of measuring the mechanical properties of the nucleus and classifying the primary structures dictating nuclear mechanics, physical plasticity of the nucleus, the emerging role of the linker of nucleoskeleton and cytoskeleton (LINC) complex in coupling the nucleus to the cytoplasm and driving the behavior of individual cells and multicellular assemblies, and the computational models currently in use to investigate the mechanisms of gene expression and cell signaling. Ongoing questions and controversies, along with promising future directions, are also discussed.

Contact guidance as a consequence of coupled morphological evolution and motility of adherent cells

Adherent cells seeded on substrates spread and evolve their morphology while simultaneously displaying motility. Phenomena such as contact guidance, viz. the alignment of cells on patterned substrates, are strongly linked to the coupling of morphological evolution with motility. Here, we employ a recently developed statistical thermodynamics framework for modelling the non-thermal fluctuating response of cells to probe this coupling. This thermodynamic framework is first extended via a Langevin style model to predict temporal responses of cells to unpatterned and patterned substrates. The Langevin model is then shown to not only predict the different experimentally observed temporal scales for morphological observables such as cell area and elongation but also the interplay of morphology with motility that ultimately leads to contact guidance.

Glycosaminoglycans modulate long-range mechanical communication between cells in collagen networks

Cells can sense and respond to mechanical forces in fibrous extracellular matrices (ECMs) over distances much greater than their size. This phenomenon, termed long-range force transmission, is enabled by the realignment (buckling) of collagen fibers along directions where the forces are tensile (compressive). However, whether other key structural components of the ECM, in particular glycosaminoglycans (GAGs), can affect the efficiency of cellular force transmission remains unclear. Here we developed a theoretical model of force transmission in collagen networks with interpenetrating GAGs, capturing the competition between tension-driven collagen fiber alignment and the swelling pressure induced by GAGs. Using this model, we show that the swelling pressure provided by GAGs increases the stiffness of the collagen network by stretching the fibers in an isotropic manner. We found that the GAG-induced swelling pressure can help collagen fibers resist buckling as the cells exert contractile forces. This mechanism impedes the alignment of collagen fibers and decreases long-range cellular mechanical communication. We experimentally validated the theoretical predictions by comparing the intensity of collagen fiber alignment between cellular spheroids cultured on collagen gels versus collagen–GAG cogels. We found significantly lower intensities of aligned collagen in collagen–GAG cogels, consistent with the prediction that GAGs can prevent collagen fiber alignment. The role of GAGs in modulating force transmission uncovered in this work can be extended to understand pathological processes such as the formation of fibrotic scars and cancer metastasis, where cells communicate in the presence of abnormally high concentrations of GAGs.

Viscoelasticity Acts as a Marker for Tumor Extracellular Matrix Characteristics

Biological materials such as extracellular matrix scaffolds, cancer cells, and tissues are often assumed to respond elastically for simplicity; the viscoelastic response is quite commonly ignored. Extracellular matrix mechanics including the viscoelasticity has turned out to be a key feature of cellular behavior and the entire shape and function of healthy and diseased tissues, such as cancer. The interference of cells with their local microenvironment and the interaction among different cell types relies both on the mechanical phenotype of each involved element. However, there is still not yet clearly understood how viscoelasticity alters the functional phenotype of the tumor extracellular matrix environment. Especially the biophysical technologies are still under ongoing improvement and further development. In addition, the effect of matrix mechanics in the progression of cancer is the subject of discussion. Hence, the topic of this review is especially attractive to collect the existing endeavors to characterize the viscoelastic features of tumor extracellular matrices and to briefly highlight the present frontiers in cancer progression and escape of cancers from therapy. Finally, this review article illustrates the importance of the tumor extracellular matrix mechano-phenotype, including the phenomenon viscoelasticity in identifying, characterizing, and treating specific cancer types.

Modelling Nuclear Morphology and Shape Transformation: A Review

As one of the most important cellular compartments, the nucleus contains genetic materials and separates them from the cytoplasm with the nuclear envelope (NE), a thin membrane that is susceptible to deformations caused by intracellular forces. Interestingly, accumulating evidence has also indicated that the morphology change of NE is tightly related to nuclear mechanotransduction and the pathogenesis of diseases such as cancer and Hutchinson–Gilford Progeria Syndrome. Theoretically, with the help of well-designed experiments, significant progress has been made in understanding the physical mechanisms behind nuclear shape transformation in different cellular processes as well as its biological implications. Here, we review different continuum-level (i.e., energy minimization, boundary integral and finite element-based) approaches that have been developed to predict the morphology and shape change of the cell nucleus. Essential gradients, relative advantages and limitations of each model will be discussed in detail, with the hope of sparking a greater research interest in this important topic in the future.

Recursive feedback between matrix dissipation and chemo-mechanical signaling drives oscillatory growth of cancer cell invadopodia

Most extracellular matrices (ECMs) are known to be dissipative, exhibiting viscoelastic and often plastic behaviors. However, the influence of dissipation, in particular mechanical plasticity in 3D confining microenvironments, on cell motility is not clear. In this study, we develop a chemo-mechanical model for dynamics of invadopodia, the protrusive structures that cancer cells use to facilitate invasion, by considering myosin recruitment, actin polymerization, matrix deformation, and mechano-sensitive signaling pathways. We demonstrate that matrix dissipation facilitates invadopodia growth by softening ECMs over repeated cycles, during which plastic deformation accumulates via cyclic ratcheting. Our model reveals that distinct protrusion patterns, oscillatory or monotonic, emerge from the interplay of timescales for polymerization-associated extension and myosin recruitment dynamics. Our model predicts the changes in invadopodia dynamics upon inhibition of myosin, adhesions, and the Rho-Rho-associated kinase (ROCK) pathway. Altogether, our work highlights the role of matrix plasticity in invadopodia dynamics and can help design dissipative biomaterials to modulate cancer cell motility.

Surface-directed engineering of tissue anisotropy in microphysiological models of musculoskeletal tissue

Here, we present an approach to model and adapt the mechanical regulation of morphogenesis that uses contractile cells as sculptors of engineered tissue anisotropy in vitro. Our method uses heterobifunctional cross-linkers to create mechanical boundary constraints that guide surface-directed sculpting of cell-laden extracellular matrix hydrogel constructs. Using this approach, we engineered linearly aligned tissues with structural and mechanical anisotropy. A multiscale in silico model of the sculpting process was developed to reveal that cell contractility increases as a function of principal stress polarization in anisotropic tissues. We also show that the anisotropic biophysical microenvironment of linearly aligned tissues potentiates soluble factor-mediated tenogenic and myogenic differentiation of mesenchymal stem cells. The application of our method is demonstrated by (i) skeletal muscle arrays to screen therapeutic modulators of acute oxidative injury and (ii) a 3D microphysiological model of lung cancer cachexia to study inflammatory and oxidative muscle injury induced by tumor-derived signals.

Computational models of migration modes improve our understanding of metastasis

Tumor cells migrate through changing microenvironments of diseased and healthy tissue, making their migration particularly challenging to describe. To better understand this process, computational models have been developed for both the ameboid and mesenchymal modes of cell migration. Here, we review various approaches that have been used to account for the physical environment's effect on cell migration in computational models, with a focus on their application to understanding cancer metastasis and the related phenomenon of durotaxis. We then discuss how mesenchymal migration models typically simulate complex cell–extracellular matrix (ECM) interactions, while ameboid migration models use a cell-focused approach that largely ignores ECM when not acting as a physical barrier. This approach greatly simplifies or ignores the mechanosensing ability of ameboid migrating cells and should be reevaluated in future models. We conclude by describing future model elements that have not been included to date but would enhance model accuracy.

The Effects of Stiffness, Fluid Viscosity, and Geometry of Microenvironment in Homeostasis, Aging, and Diseases: A Brief Review

Cells sense biophysical cues in the micro-environment and respond to the cues biochemically and biophysically. Proper responses from cells are critical to maintain the homeostasis in the body. Abnormal biophysical cues will cause pathological development in the cells; pathological or aging cells, on the other hand, can alter their micro-environment to become abnormal. In this minireview, we discuss four important biophysical cues of the micro-environment-stiffness, curvature, extracellular matrix (ECM) architecture and viscosity-in terms of their roles in health, aging, and diseases.

Cell Shape and Durotaxis Explained from Cell-Extracellular Matrix Forces and Focal Adhesion Dynamics

Many cells are small and rounded on soft extracellular matrices (ECM), elongated on stiffer ECMs, and flattened on hard ECMs. Cells also migrate up stiffness gradients (durotaxis). Using a hybrid cellular Potts and finite-element model extended with ODE-based models of focal adhesion (FA) turnover, we show that the full range of cell shape and durotaxis can be explained in unison from dynamics of FAs, in contrast to previous mathematical models. In our 2D cell-shape model, FAs grow due to cell traction forces. Forces develop faster on stiff ECMs, causing FAs to stabilize and, consequently, cells to spread on stiff ECMs. If ECM stress further stabilizes FAs, cells elongate on substrates of intermediate stiffness. We show that durotaxis follows from the same set of assumptions. Our model contributes to the understanding of the basic responses of cells to ECM stiffness, paving the way for future modeling of more complex cell-ECM interactions.

Collagen microarchitecture mechanically controls myofibroblast differentiation

Altered microarchitecture of collagen type I is a hallmark of wound healing and cancer that is commonly attributed to myofibroblasts. However, it remains unknown which effect collagen microarchitecture has on myofibroblast differentiation. Here, we combined experimental and computational approaches to investigate the hypothesis that the microarchitecture of fibrillar collagen networks mechanically regulates myofibroblast differentiation of adipose stromal cells (ASCs) independent of bulk stiffness. Collagen gels with controlled fiber thickness and pore size were microfabricated by adjusting the gelation temperature while keeping their concentration constant. Rheological characterization and simulation data indicated that networks with thicker fibers and larger pores exhibited increased strain-stiffening relative to networks with thinner fibers and smaller pores. Accordingly, ASCs cultured in scaffolds with thicker fibers were more contractile, expressed myofibroblast markers, and deposited more extended fibronectin fibers. Consistent with elevated myofibroblast differentiation, ASCs in scaffolds with thicker fibers exhibited a more proangiogenic phenotype that promoted endothelial sprouting in a contractility-dependent manner. Our findings suggest that changes of collagen microarchitecture regulate myofibroblast differentiation and fibrosis independent of collagen quantity and bulk stiffness by locally modulating cellular mechanosignaling. These findings have implications for regenerative medicine and anticancer treatments.

A biomechanical model for the transendothelial migration of cancer cells

We propose a biomechanical model for the extravasation of a tumor cell (TC) through the endothelium of a blood vessel. Based on prior in vitro observations, we assume that the TC extends a protrusion between adjacent endothelial cells (ECs) that adheres to the basement membrane via focal adhesions (FAs). As the protrusion grows in size and branches out, the actomyosin contraction along the stress fibers (SFs) inside the protrusion pulls the relatively rigid nucleus through the endothelial opening. We model the chemo-mechanics of the SFs and the FAs by following the kinetics of the active myosin motors and high-affinity integrins, subject to mechanical feedback. This is incorporated into a finite-element simulation of the extravasation process, with the contractile force pulling the nucleus of the TC against elastic resistance of the ECs. To account for the interaction between the TC nucleus and the endothelium, we consider two scenarios: solid-solid contact and lubrication by cytosol. The former gives a lower bound for the required contractile force to realize transmigration, while the latter provides a more realistic representation of the process. Using physiologically reasonable parameters, our model shows that the SF and FA ensemble can produce a contractile force on the order of 70 nN, which is sufficient to deform the ECs and enable transmigration. Furthermore, we use an atomic force microscope to measure the resistant force on a human bladder cancer cell that is pushed through an endothelium cultured in vitro. The magnitude of the required force turns out to be in the range of 70-100 nN, comparable to the model predictions.

Myofibroblast Gene Expression Profile after Tooth Extraction in the Rabbit

After tooth extraction, the alveolar bone tends to shrink in volume, especially on the vestibular side. The role of myofibroblasts in bone remodeling has not been sufficiently investigated. The aim of the present study was to explore the gene expression related to myofibroblasts presence and activity during a 90-day healing period after tooth extraction. The study included 36 rabbits, and a single tooth extraction was performed on each rabbit. The extractive sockets were randomly distributed to natural healing or to scarification of the wound. The sacrifices were staggered in such a manner that animals contributed with sockets representing 2, 7, 15, 30, 60, and 90 days of healing. Nanostring technology was used to evaluate the expression of a wide panel consisting in 148 genes related to the activation, induction, and suppression of myofibroblasts, socket microenvironment, and autophagy. We found that the expression profile of this custom panel was time-related. The post-extractive socket was subjected to significant gene expression changes after 15 days: the genes involved in the induction of myofibroblasts were up-regulated in the first 15-day period and down-regulated during the rest of the follow-up. The study suggested that myofibroblasts play a major role in the immediate 15-day period following tooth extraction.

Force chains in cell-cell mechanical communication

Force chains (FCs) are a key determinant of the micromechanical properties and behaviour of heterogeneous materials, such as granular systems. However, less is known about FCs in fibrous materials, such as the networks composing the extracellular matrix (ECM) of biological systems. Using a finite-element computational model, we simulated the contraction of a single cell and two nearby cells embedded in two-dimensional fibrous elastic networks and analysed the tensile FCs that developed in the ECM. The role of ECM nonlinear elasticity on FC formation was evaluated by considering linear and nonlinear, i.e. exhibiting 'buckling' and/or 'strain-stiffening', stress-strain curves. The effect of the degree of cell contraction and network coordination value was assessed. We found that nonlinear elasticity of the ECM fibres influenced the structure of the FCs, facilitating the transition towards more distinct chains that were less branched and more radially oriented than the chains formed in linear elastic networks. When two neighbouring cells contract, a larger number of FCs bridged between the cells in nonlinear networks, and these chains had a larger effective rigidity than the chains that did not reach a neighbouring cell. These results suggest that FCs function as a route for mechanical communication between distant cells and highlight the contribution of ECM fibre nonlinear elasticity to the formation of FCs.

Rigidity controls human desmoplastic matrix anisotropy to enable pancreatic cancer cell spread via extracellular signal-regulated kinase 2

It is predicted that pancreatic ductal adenocarcinoma (PDAC) will become the second most lethal cancer in the US by 2030. PDAC includes a fibrous-like stroma, desmoplasia, encompassing most of the tumor mass, which is produced by cancer-associated fibroblasts (CAFs) and includes their cell-derived extracellular matrices (CDMs). Since elimination of desmoplasia has proven detrimental to patients, CDM reprogramming, as opposed to stromal ablation, is therapeutically desirable. Hence, efforts are being made to harness desmoplasia's anti-tumor functions. We conducted biomechanical manipulations, using variations of pathological and physiological substrates in vitro, to culture patient-harvested CAFs and generate CDMs that restrict PDAC growth and spread. We posited that extrinsic modulation of the environment, via substrate rigidity, influences CAF's cell-intrinsic forces affecting CDM production. Substrates used were polyacrylamide gels of physiological (~1.5 kPa) or pathological (~7 kPa) stiffnesses. Results showed that physiological substrates influenced CAFs to generate CDMs similar to normal/control fibroblasts. We found CDMs to be softer than the corresponding underlying substrates, and CDM fiber anisotropy (i.e., alignment) to be biphasic and informed via substrate-imparted morphological CAF aspect ratios. The biphasic nature of CDM fiber anisotropy was mathematically modeled and proposed a correlation between CAF aspect ratios and CDM alignment; regulated by extrinsic and intrinsic forces to conserve minimal free energy. Biomechanical manipulation of CDMs, generated on physiologically soft substrates, leads to reduction in nuclear translocation of pERK1/2 in KRAS mutated pancreatic cells. ERK2 was found essential for CDM-regulated tumor cell spread. In vitro findings correlated with in vivo observations; nuclear pERK1/2 is significantly high in human PDAC samples. The study suggests that altering underlying substrates enable CAFs to remodel CDMs and restrict pancreatic cancer cell spread in an ERK2 dependent manner.

An electro-osmotic microfluidic system to characterize cancer cell migration under confinement

We have developed a novel electro-osmotic microfluidic system to apply precisely controlled osmolarity gradients to cancer cells in micro-channels. We observed that albeit adhesion is not required for cells to migrate in such a confined microenvironment, the migrating velocity of cells is strongly influenced by the interactions between the cells and the channel wall, with a stronger adhesion leading to diminished cell motility. Furthermore, through examining more than 20 different types of cancer cells, we found a linear positive correlation between the protein concentration of the aquaporin-4 (AQP4) and the cell migrating speed. Knockdown of AQP4 in invasive re-populated cancer stem cells reduced their migration capability down to the level that is comparable to their parental cancer cells. Interestingly, these observations can all be quantitatively explained by the osmotic engine model where the cell movement is assumed to be driven by cross-membrane ion/water transport, while adhesion acts as a frictional resistance against the cell motility. By providing versatile and controllable features in regulating and characterizing the migration capability of cells, our system may serve as a useful tool in quantifying how cell motility is influenced by different physical and biochemical factors, as well as elucidating the mechanisms behind, in the future.

Cell motility, contact guidance, and durotaxis

Mechanical properties of the substrate play a vital role in cell motility. In particular, cells have been shown to migrate along aligned fibers in the substrate (contact guidance) and up stiffness gradients (durotaxis). Here we present a simple mechanical model for cell migration coupled to substrate properties, by placing a simulated cell on a lattice mimicking biopolymer gels or hydrogels. In our model cells attach to the substrate via focal adhesions (FAs). As the cells contract, forces are generated at the FAs, determining their maturation and detachment. At the same time, the cell was also allowed to move and rotate to maintain force and torque balance. Our model, in which the cells only have access to information regarding forces acting at the FAs, without a prior knowledge of the substrate stiffness or geometry, is able to reproduce both contact guidance and durotaxis.

Elastic interactions between anisotropically contracting circular cells

We study interactions between biological cells that apply anisotropic active mechanical forces on an elastic substrate. We model the cells as thin disks that along their perimeters apply radial, but angle-dependent forces on the substrate. We obtain analytical expressions for the elastic energy stored in the substrate as a function of the distance between the cells, the Fourier modes of applied forces, and their phase angles. We show how the relative phases of the forces applied by the cells can switch the interaction between attractive and repulsive, and relate our results to those for linear force dipoles. For long enough distances, the interaction energy decays in magnitude as a power law of the cell-cell distance with an integer exponent that generally increases with the Fourier modes of the applied forces.

Dynamic fibroblast contractions attract remote macrophages in fibrillar collagen matrix

Macrophage (Mϕ)-fibroblast interactions coordinate tissue repair after injury whereas miscommunications can result in pathological healing and fibrosis. We show that contracting fibroblasts generate deformation fields in fibrillar collagen matrix that provide far-reaching physical cues for Mϕ. Within collagen deformation fields created by fibroblasts or actuated microneedles, Mϕ migrate towards the force source from several hundreds of micrometers away. The presence of a dynamic force source in the matrix is critical to initiate and direct Mϕ migration. In contrast, collagen condensation and fiber alignment resulting from fibroblast remodelling activities or chemotactic signals are neither required nor sufficient to guide Mϕ migration. Binding of α2β1 integrin and stretch-activated channels mediate Mϕ migration and mechanosensing in fibrillar collagen ECM. We propose that Mϕ mechanosense the velocity of local displacements of their substrate, allowing contractile fibroblasts to attract Mϕ over distances that exceed the range of chemotactic gradients.

Macrophages play an important role in wound healing but the guidance cues driving macrophages to sites of repair are still not clear. Here the authors discover that macrophages are attracted to contracting fibroblasts by responding to locally sensed displacements of collagen fibres.

Entropic Forces Drive Cellular Contact Guidance

Contact guidance-the widely known phenomenon of cell alignment induced by anisotropic environmental features-is an essential step in the organization of adherent cells, but the mechanisms by which cells achieve this orientational ordering remain unclear. Here, we seeded myofibroblasts on substrates micropatterned with stripes of fibronectin and observed that contact guidance emerges at stripe widths much greater than the cell size. To understand the origins of this surprising observation, we combined morphometric analysis of cells and their subcellular components with a, to our knowledge, novel statistical framework for modeling nonthermal fluctuations of living cells. This modeling framework is shown to predict not only the trends but also the statistical variability of a wide range of biological observables, including cell (and nucleus) shapes, sizes, and orientations, as well as stress-fiber arrangements within the cells with remarkable fidelity with a single set of cell parameters. By comparing observations and theory, we identified two regimes of contact guidance: 1) guidance on stripe widths smaller than the cell size (w ≤ 160 μm), which is accompanied by biochemical changes within the cells, including increasing stress-fiber polarization and cell elongation; and 2) entropic guidance on larger stripe widths, which is governed by fluctuations in the cell morphology. Overall, our findings suggest an entropy-mediated mechanism for contact guidance associated with the tendency of cells to maximize their morphological entropy through shape fluctuations.

Remodeling of the Collagen Matrix in Aging Skin Promotes Melanoma Metastasis and Affects Immune Cell Motility

Physical changes in skin are among the most visible signs of aging. We found that young dermal fibroblasts secrete high levels of extracellular matrix (ECM) constituents, including proteoglycans, glycoproteins, and cartilage-linking proteins. The most abundantly secreted was HAPLN1, a hyaluronic and proteoglycan link protein. HAPLN1 was lost in aged fibroblasts, resulting in a more aligned ECM that promoted metastasis of melanoma cells. Reconstituting HAPLN1 inhibited metastasis in an aged microenvironment, in 3-D skin reconstruction models, and in vivo. Intriguingly, aged fibroblast-derived matrices had the opposite effect on the migration of T cells, inhibiting their motility. HAPLN1 treatment of aged fibroblasts restored motility of mononuclear immune cells, while impeding that of polymorphonuclear immune cells, which in turn affected regulatory T-cell recruitment. These data suggest that although age-related physical changes in the ECM can promote tumor cell motility, they may adversely affect the motility of some immune cells, resulting in an overall change in the immune microenvironment. Understanding the physical changes in aging skin may provide avenues for more effective therapy for older patients with melanoma. SIGNIFICANCE: These data shed light on the mechanochemical interactions that occur between aged skin, tumor, and immune cell populations, which may affect tumor metastasis and immune cell infiltration, with implications for the efficacy of current therapies for melanoma.See related commentary by Marie and Merlino, p. 19.This article is highlighted in the In This Issue feature, p. 1.

Thermodynamic Modeling of the Statistics of Cell Spreading on Ligand-Coated Elastic Substrates

Biological spread cells exist in a perpetually fluctuating state and therefore cannot be described in terms of a unique deterministic system. For modeling approaches to provide novel insight and uncover new mechanisms that drive cell behavior, a framework is required that progresses from traditional deterministic methods (whereby simulation of an experiment predicts a single outcome). In this study, we implement a new, to our knowledge, modeling approach for the analysis of cell spreading on ligand-coated substrates, extending the framework for nonequilibrium thermodynamics of cells developed by Shishvan et al. to include active focal adhesion assembly. We demonstrate that the model correctly predicts the coupled influence of surface collagen density and substrate stiffness on cell spreading, as reported experimentally by Engler et al. Low surface collagen densities are shown to result in a high probability that cells will be restricted to low spread areas. Furthermore, elastic free energy induced by substrate deformation lowers the probability of observing a highly spread cell, and, consequentially, lower cell tractions affect the assembly of focal adhesions. Experimentally measurable observables such as cell spread area and aspect ratio can be directly postprocessed from the computed homeostatic ensemble of (several million) spread states. This allows for the prediction of mean and SDs of such experimental observables. This class of cell mechanics modeling presents a significant advance on conventional deterministic approaches.

The homeostatic ensemble for cells

Cells are quintessential examples of out-of-equilibrium systems, but they maintain a homeostatic state over a timescale of hours to days. As a consequence, the statistics of all observables is remarkably consistent. Here, we develop a statistical mechanics framework for living cells by including the homeostatic constraint that exists over the interphase period of the cell cycle. The consequence is the introduction of the concept of a homeostatic ensemble and an associated homeostatic temperature, along with a formalism for the (dynamic) homeostatic equilibrium that intervenes to allow living cells to evade thermodynamic decay. As a first application, the framework is shown to accurately predict the observed effect of the mechanical environment on the in vitro response of smooth muscle cells. This includes predictions that both the mean values and diversity/variability in the measured values of observables such as cell area, shape and tractions decrease with decreasing stiffness of the environment. Thus, we argue that the observed variabilities are inherent to the entropic nature of the homeostatic equilibrium of cells and not a result of in vitro experimental errors.

Matching material and cellular timescales maximizes cell spreading on viscoelastic substrates

Recent evidence has shown that, in addition to rigidity, the viscous response of the extracellular matrix (ECM) significantly affects the behavior and function of cells. However, the mechanism behind such mechanosensitivity toward viscoelasticity remains unclear. In this study, we systematically examined the dynamics of motor clutches (i.e., focal adhesions) formed between the cell and a viscoelastic substrate using analytical methods and direct Monte Carlo simulation. Interestingly, we observe that, for low ECM rigidity, maximum cell spreading is achieved at an optimal level of viscosity in which the substrate relaxation time falls between the timescale for clutch binding and its characteristic binding lifetime. That is, viscosity serves to stiffen soft substrates on a timescale faster than the clutch off-rate, which enhances cell-ECM adhesion and cell spreading. On the other hand, for substrates that are stiff, our model predicts that viscosity will not influence cell spreading, since the bound clutches are saturated by the elevated stiffness. The model was tested and validated using experimental measurements on three different material systems and explained the different observed effects of viscosity on each substrate. By capturing the mechanism by which substrate viscoelasticity affects cell spreading across a wide range of material parameters, our analytical model provides a useful tool for designing biomaterials that optimize cellular adhesion and mechanosensing.

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.

Functional and Biomimetic Materials for Engineering of the Three-Dimensional Cell Microenvironment

The cell microenvironment has emerged as a key determinant of cell behavior and function in development, physiology, and pathophysiology. The extracellular matrix (ECM) within the cell microenvironment serves not only as a structural foundation for cells but also as a source of three-dimensional (3D) biochemical and biophysical cues that trigger and regulate cell behaviors. Increasing evidence suggests that the 3D character of the microenvironment is required for development of many critical cell responses observed in vivo, fueling a surge in the development of functional and biomimetic materials for engineering the 3D cell microenvironment. Progress in the design of such materials has improved control of cell behaviors in 3D and advanced the fields of tissue regeneration, in vitro tissue models, large-scale cell differentiation, immunotherapy, and gene therapy. However, the field is still in its infancy, and discoveries about the nature of cell-microenvironment interactions continue to overturn much early progress in the field. Key challenges continue to be dissecting the roles of chemistry, structure, mechanics, and electrophysiology in the cell microenvironment, and understanding and harnessing the roles of periodicity and drift in these factors. This review encapsulates where recent advances appear to leave the ever-shifting state of the art, and it highlights areas in which substantial potential and uncertainty remain.

Interplay of Platelet Contractility and Elasticity of Fibrin/Erythrocytes in Blood Clot Retraction

Blood clot contraction (retraction) is driven by platelet-generated forces propagated by the fibrin network and results in clot shrinkage and deformation of erythrocytes. To elucidate the mechanical nature of this process, we developed a model that combines an active contractile motor element with passive viscoelastic elements. Despite its importance for thrombosis and wound healing, clot contraction is poorly understood. This model predicts how clot contraction occurs due to active contractile platelets interacting with a viscoelastic material, rather than to the poroelastic nature of fibrin, and explains the observed dynamics of clot size, ultrastructure, and measured forces. Mechanically passive erythrocytes and fibrin are present in series and parallel to active contractile cells. This mechanical interplay induces compressive and tensile resistance, resulting in increased contractile force and a reduced extent of contraction in the presence of erythrocytes. This experimentally validated model provides the fundamental mechanical basis for understanding contraction of blood clots.

A Chemomechanical Model for Nuclear Morphology and Stresses during Cell Transendothelial Migration

It is now evident that the cell nucleus undergoes dramatic shape changes during important cellular processes such as cell transmigration through extracellular matrix and endothelium. Recent experimental data suggest that during cell transmigration the deformability of the nucleus could be a limiting factor, and the morphological and structural alterations that the nucleus encounters can perturb genomic organization that in turn influences cellular behavior. Despite its importance, a biophysical model that connects the experimentally observed nuclear morphological changes to the underlying biophysical factors during transmigration through small constrictions is still lacking. Here, we developed a universal chemomechanical model that describes nuclear strains and shapes and predicts thresholds for the rupture of the nuclear envelope and for nuclear plastic deformation during transmigration through small constrictions. The model includes actin contraction and cytosolic back pressure that squeeze the nucleus through constrictions and overcome the mechanical resistance from deformation of the nucleus and the constrictions. The nucleus is treated as an elastic shell encompassing a poroelastic material representing the nuclear envelope and inner nucleoplasm, respectively. Tuning the chemomechanical parameters of different components such as cell contractility and nuclear and matrix stiffnesses, our model predicts the lower bounds of constriction size for successful transmigration. Furthermore, treating the chromatin as a plastic material, our model faithfully reproduced the experimentally observed irreversible nuclear deformations after transmigration in lamin-A/C-deficient cells, whereas the wild-type cells show much less plastic deformation. Along with making testable predictions, which are in accord with our experiments and existing literature, our work provides a realistic framework to assess the biophysical modulators of nuclear deformation during cell transmigration.

Free energy analysis of cell spreading

In this study we present a steady-state adaptation of the thermodynamically motivated stress fiber (SF) model of Vigliotti et al. (2015). We implement this steady-state formulation in a non-local finite element setting where we also consider global conservation of the total number of cytoskeletal proteins within the cell, global conservation of the number of binding integrins on the cell membrane, and adhesion limiting ligand density on the substrate surface. We present a number of simulations of cell spreading in which we consider a limited subset of the possible deformed spread-states assumed by the cell in order to examine the hypothesis that free energy minimization drives the process of cell spreading. Simulations suggest that cell spreading can be viewed as a competition between (i) decreasing cytoskeletal free energy due to strain induced assembly of cytoskeletal proteins into contractile SFs, and (ii) increasing elastic free energy due to stretching of the mechanically passive components of the cell. The computed minimum free energy spread area is shown to be lower for a cell on a compliant substrate than on a rigid substrate. Furthermore, a low substrate ligand density is found to limit cell spreading. The predicted dependence of cell spread area on substrate stiffness and ligand density is in agreement with the experiments of Engler et al. (2003). We also simulate the experiments of Théry et al. (2006), whereby initially circular cells deform and adhere to "V-shaped" and "Y-shaped" ligand patches. Analysis of a number of different spread states reveals that deformed configurations with the lowest free energy exhibit a SF distribution that corresponds to experimental observations, i.e. a high concentration of highly aligned SFs occurs along free edges, with lower SF concentrations in the interior of the cell. In summary, the results of this study suggest that cell spreading is driven by free energy minimization based on a competition between decreasing cytoskeletal free energy and increasing passive elastic free energy.

Multiscale model predicts increasing focal adhesion size with decreasing stiffness in fibrous matrices

We describe a multiscale model that incorporates force-dependent mechanical plasticity induced by interfiber cross-link breakage and stiffness-dependent cellular contractility to predict focal adhesion (FA) growth and mechanosensing in fibrous extracellular matrices (ECMs). The model predicts that FA size depends on both the stiffness of ECM and the density of ligands available to form adhesions. Although these two quantities are independent in commonly used hydrogels, contractile cells break cross-links in soft fibrous matrices leading to recruitment of fibers, which increases the ligand density in the vicinity of cells. Consequently, although the size of focal adhesions increases with ECM stiffness in nonfibrous and elastic hydrogels, plasticity of fibrous networks leads to a departure from the well-described positive correlation between stiffness and FA size. We predict a phase diagram that describes nonmonotonic behavior of FA in the space spanned by ECM stiffness and recruitment index, which describes the ability of cells to break cross-links and recruit fibers. The predicted decrease in FA size with increasing ECM stiffness is in excellent agreement with recent observations of cell spreading on electrospun fiber networks with tunable cross-link strengths and mechanics. Our model provides a framework to analyze cell mechanosensing in nonlinear and inelastic ECMs.

Modeling the two-way feedback between contractility and matrix realignment reveals a nonlinear mode of cancer cell invasion

Cancer cell invasion from primary tumors is mediated by a complex interplay between cellular adhesions, actomyosin-driven contractility, and the physical characteristics of the extracellular matrix (ECM). Here, we incorporate a mechanochemical free-energy-based approach to elucidate how the two-way feedback loop between cell contractility (induced by the activity of chemomechanical interactions such as Ca2+ and Rho signaling pathways) and matrix fiber realignment and strain stiffening enables the cells to polarize and develop contractile forces to break free from the tumor spheroids and invade into the ECM. Interestingly, through this computational model, we are able to identify a critical stiffness that is required by the matrix to break intercellular adhesions and initiate cell invasion. Also, by considering the kinetics of the cell movement, our model predicts a biphasic invasiveness with respect to the stiffness of the matrix. These predictions are validated by analyzing the invasion of melanoma cells in collagen matrices of varying concentration. Our model also predicts a positive correlation between the elongated morphology of the invading cells and the alignment of fibers in the matrix, suggesting that cell polarization is directly proportional to the stiffness and alignment of the matrix. In contrast, cells in nonfibrous matrices are found to be rounded and not polarized, underscoring the key role played by the nonlinear mechanics of fibrous matrices. Importantly, our model shows that mechanical principles mediated by the contractility of the cells and the nonlinearity of the ECM behavior play a crucial role in determining the phenotype of the cell invasion.

Fibroblasts Slow Conduction Velocity in a Reconstituted Tissue Model of Fibrotic Cardiomyopathy

Myocardial function deteriorates over the course of fibrotic cardiomyopathy, due to electrophysiological and mechanical effects of myofibroblasts that are not completely understood. Although a range of experimental model systems and associated theoretical treatments exist at the levels of isolated cardiomyocytes and planar co-cultures of myofibroblasts and cardiomyocytes, interactions between these cell types at the tissue level are less clear. We studied these interactions through an engineered heart tissue (EHT) model of fibrotic myocardium and a mathematical model of the effects of cellular composition on EHT impulse conduction velocity. The EHT model allowed for modulation of cardiomyocyte and myofibroblast volume fractions, and observation of cell behavior in a three-dimensional environment that is more similar to native heart tissue than is planar cell culture. The cardiomyocyte and myofibroblast volume fractions determined the retardation of impulse conduction (spread of the action potential) in EHTs as measured by changes of the fluorescence of the Ca²⁺ probe, Fluo-2. Interpretation through our model showed retardation far in excess of predictions by homogenization theory, with conduction ceasing far below the fibroblast volume fraction associated with steric percolation. Results point to an important multiscale structural role of myofibroblasts in attenuating impulse conduction in fibrotic cardiomyopathy.