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This study reports novel findings that link E-cadherin (also known as CDH1)-mediated force-transduction signaling to vinculin targeting to intercellular junctions via epidermal growth factor receptor (EGFR) and integrins. These results build on previous findings that demonstrated that mechanically perturbed E-cadherin receptors activate phosphoinositide 3-kinase and downstream integrins in an EGFR-dependent manner. Results of this study show that this EGFR-mediated kinase cascade controls the force-dependent recruitment of vinculin to stressed E-cadherin complexes - a key early signature of cadherin-based mechanotransduction. Vinculin targeting requires its phosphorylation at tyrosine 822 by Abl family kinases (hereafter Abl), but the origin of force-dependent Abl activation had not been identified. We now present evidence that integrin activation, which is downstream of EGFR signaling, controls Abl activation, thus linking E-cadherin to Abl through a mechanosensitive signaling network. These findings place EGFR and integrins at the center of a positive-feedback loop, through which force-activated E-cadherin signals regulate vinculin recruitment to cadherin complexes in response to increased intercellular tension.This article has an associated First Person interview with the first author of the paper.
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Cadherinas/metabolismo , Receptores ErbB/metabolismo , Integrinas/metabolismo , Uniones Intercelulares/metabolismo , Vinculina/química , Vinculina/metabolismo , Cadherinas/genética , Línea Celular Tumoral , Receptores ErbB/genética , Humanos , Integrinas/genética , Uniones Intercelulares/genética , Mecanotransducción Celular , Fosfatidilinositol 3-Quinasas/genética , Fosfatidilinositol 3-Quinasas/metabolismo , Unión Proteica , Vinculina/genéticaRESUMEN
For an organism to develop and maintain homeostasis, cell types with distinct functions must often be separated by physical boundaries. The formation and maintenance of such boundaries are commonly attributed to mechanisms restricted to the cells lining the boundary. Here we show that, besides these local subcellular mechanisms, the formation and maintenance of tissue boundaries involves long-lived, long-ranged mechanical events. Following contact between two epithelial monolayers expressing, respectively, EphB2 and its ligand ephrinB1, both monolayers exhibit oscillatory patterns of traction forces and intercellular stresses that tend to pull cell-matrix adhesions away from the boundary. With time, monolayers jam, accompanied by the emergence of deformation waves that propagate away from the boundary. This phenomenon is not specific to EphB2/ephrinB1 repulsion but is also present during the formation of boundaries with an inert interface and during fusion of homotypic epithelial layers. Our findings thus unveil a global physical mechanism that sustains tissue separation independently of the biochemical and mechanical features of the local tissue boundary.
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Relojes Biológicos , Efrina-B1/metabolismo , Células Epiteliales/metabolismo , Matriz Extracelular/metabolismo , Receptor EphB2/metabolismo , Estrés Fisiológico , Animales , Perros , Efrina-B1/genética , Células Epiteliales/citología , Epitelio/metabolismo , Matriz Extracelular/genética , Células de Riñón Canino Madin Darby , Receptor EphB2/genéticaRESUMEN
Tissue rigidity regulates processes in development, cancer and wound healing. However, how cells detect rigidity, and thereby modulate their behaviour, remains unknown. Here, we show that sensing and adaptation to matrix rigidity in breast myoepithelial cells is determined by the bond dynamics of different integrin types. Cell binding to fibronectin through either α5ß1 integrins (constitutively expressed) or αvß6 integrins (selectively expressed in cancer and development) adapts force generation, actin flow and integrin recruitment to rigidities associated with healthy or malignant tissue, respectively. In vitro experiments and theoretical modelling further demonstrate that this behaviour is explained by the different binding and unbinding rates of both integrin types to fibronectin. Moreover, rigidity sensing through differences in integrin bond dynamics applies both when integrins bind separately and when they compete for binding to fibronectin.
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Antígenos de Neoplasias/metabolismo , Fibronectinas/metabolismo , Integrinas/metabolismo , Mecanotransducción Celular/fisiología , Modelos Biológicos , Receptores de Vitronectina/metabolismo , Antígenos de Neoplasias/genética , Células Cultivadas , Fibronectinas/genética , Humanos , Integrinas/genética , Receptores de Vitronectina/genéticaRESUMEN
Cells isolated from their native tissues and cultured in vitro face different selection pressures than those cultured in vivo. These pressures induce a profound transformation that reshapes the cell, alters its genome, and transforms the way it senses and generates forces. In this perspective, we focus on the evidence that cells cultured on conventional polystyrene substrates display a fundamentally different mechanobiology than their in vivo counterparts. We explore the role of adhesion reinforcement in this transformation and to what extent it is reversible. We argue that this mechanoadaptation is often understood as a mechanical memory. We propose some strategies to mitigate the effects of on-plastic culture on mechanobiology, such as organoid-inspired protocols or mechanical priming. While isolating cells from their native tissues and culturing them on artificial substrates has revolutionized biomedical research, it has also transformed cellular forces. Only by understanding and controlling them, we can improve their truthfulness and validity.
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Exposure of experimental rodents to controlled cycles of light, food, and temperature is important when investigating alterations in circadian cycles that profoundly influence health and disease. However, applying such stimuli simultaneously is difficult in practice. We aimed to design, build, test, and open-source describe a simple device that subjects a conventional mouse cage to independent cycles of physiologically relevant environmental variables. The device is based on a box enclosing the rodent cage to modify the light, feeding, and temperature environments. The device provides temperature-controlled air conditioning (heating or cooling) by a Peltier module and includes programmable feeding and illumination. All functions are set by a user-friendly front panel for independent cycle programming. Bench testing with a model simulating the CO2 production of mice in the cage showed: a) suitable air renewal (by measuring actual ambient CO2), b) controlled realistic illumination at the mouse enclosure (measured by a photometer), c) stable temperature control, and d) correct cycling of light, feeding, and temperature. The cost of all the supplies (retail purchased by e-commerce) was <300 US$. Detailed technical information is open-source provided, allowing for any user to reliably reproduce or modify the device. This approach can considerably facilitate circadian research since using one of the described low-cost devices for any mouse group with a given light-food-temperature paradigm allows for all the experiments to be performed simultaneously, thereby requiring no changes in the light/temperature of a general-use laboratory.
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Since the emergence of 3D bioprinting technology, both synthetic and natural materials have been used to develop bioinks for producing cell-laden cardiac grafts. To this end, extracellular-matrix (ECM)-derived hydrogels can be used to develop scaffolds that closely mimic the complex 3D environments for cell culture. This study presents a novel cardiac bioink based on hydrogels exclusively derived from decellularized porcine myocardium loaded with human-bone-marrow-derived mesenchymal stromal cells. Hence, the hydrogel can be used to develop cell-laden cardiac patches without the need to add other biomaterials or use additional crosslinkers. The scaffold ultrastructure and mechanical properties of the bioink were characterized to optimize its production, specifically focusing on the matrix enzymatic digestion time. The cells were cultured in 3D within the developed hydrogels to assess their response. The results indicate that the hydrogels fostered inter-cell and cell-matrix crosstalk after 1 week of culture. In conclusion, the bioink developed and presented in this study holds great potential for developing cell-laden customized patches for cardiac repair.
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The function of organs such as lungs, kidneys and mammary glands relies on the three-dimensional geometry of their epithelium. To adopt shapes such as spheres, tubes and ellipsoids, epithelia generate mechanical stresses that are generally unknown. Here we engineer curved epithelial monolayers of controlled size and shape and map their state of stress. We design pressurized epithelia with circular, rectangular and ellipsoidal footprints. We develop a computational method, called curved monolayer stress microscopy, to map the stress tensor in these epithelia. This method establishes a correspondence between epithelial shape and mechanical stress without assumptions of material properties. In epithelia with spherical geometry we show that stress weakly increases with areal strain in a size-independent manner. In epithelia with rectangular and ellipsoidal cross-section we find pronounced stress anisotropies that impact cell alignment. Our approach enables a systematic study of how geometry and stress influence epithelial fate and function in three-dimensions.
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Células Epiteliales , Microscopía , Estrés Mecánico , EpitelioRESUMEN
Background: Patients with obstructive sleep apnea (OSA) experience recurrent hypoxemic events with a frequency sometimes exceeding 60 events/h. These episodic events induce downstream transient hypoxia in the parenchymal tissue of all organs, thereby eliciting the pathological consequences of OSA. Whereas experimental models currently apply intermittent hypoxia to cells conventionally cultured in 2D plates, there is no well-characterized setting that will subject cells to well-controlled intermittent hypoxia in a 3D environment and enable the study of the effects of OSA on the cells of interest while preserving the underlying tissue environment. Aim: To design and characterize an experimental approach that exposes cells to high-frequency intermittent hypoxia mimicking OSA in 3D (hydrogels or tissue slices). Methods: Hydrogels made from lung extracellular matrix (L-ECM) or brain tissue slices (300-800-µm thickness) were placed on a well whose bottom consisted of a permeable silicone membrane. The chamber beneath the membrane was subjected to a square wave of hypoxic/normoxic air. The oxygen concentration at different depths within the hydrogel/tissue slice was measured with an oxygen microsensor. Results: 3D-seeded cells could be subjected to well-controlled and realistic intermittent hypoxia patterns mimicking 60 apneas/h when cultured in L-ECM hydrogels ≈500 µm-thick or ex-vivo in brain slices 300-500 µm-thick. Conclusion: This novel approach will facilitate the investigation of the effects of intermittent hypoxia simulating OSA in 3D-residing cells within the parenchyma of different tissues/organs.
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Emerging evidence points to coordinated action of chemical and mechanical cues during brain development. At early stages of neocortical development, angiogenic factors and chemokines such as CXCL12, ephrins, and semaphorins assume crucial roles in orchestrating neuronal migration and axon elongation of postmitotic neurons. Here we explore the intrinsic mechanical properties of the developing marginal zone of the pallium in the migratory pathways and brain distribution of the pioneer Cajal-Retzius cells. These neurons are generated in several proliferative regions in the developing brain (e.g., the cortical hem and the pallial subpallial boundary) and migrate tangentially in the preplate/marginal zone covering the upper portion of the developing cortex. These cells play crucial roles in correct neocortical layer formation by secreting several molecules such as Reelin. Our results indicate that the motogenic properties of Cajal-Retzius cells and their perinatal distribution in the marginal zone are modulated by both chemical and mechanical factors, by the specific mechanical properties of Cajal-Retzius cells, and by the differential stiffness of the migratory routes. Indeed, cells originating in the cortical hem display higher migratory capacities than those generated in the pallial subpallial boundary which may be involved in the differential distribution of these cells in the dorsal-lateral axis in the developing marginal zone.
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In this Primer, Sunyer and Trepat introduce durotaxis, the mode of migration by which cells follow gradients of extracellular matrix stiffness.
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Adhesión Celular/fisiología , Movimiento Celular/fisiología , Taxia/fisiología , Animales , Fenómenos Biomecánicos , Matriz Extracelular , HumanosRESUMEN
Shape-dependent local differentials in cell proliferation are considered to be a major driving mechanism of structuring processes in vivo, such as embryogenesis, wound healing, and angiogenesis. However, the specific biophysical signaling by which changes in cell shape contribute to cell cycle regulation remains poorly understood. Here, we describe our study of the roles of nuclear volume and cytoskeletal mechanics in mediating shape control of proliferation in single endothelial cells. Micropatterned adhesive islands were used to independently control cell spreading and elongation. We show that, irrespective of elongation, nuclear volume and apparent chromatin decondensation of cells in G1 systematically increased with cell spreading and highly correlated with DNA synthesis (percent of cells in the S phase). In contrast, cell elongation dramatically affected the organization of the actin cytoskeleton, markedly reduced both cytoskeletal stiffness (measured dorsally with atomic force microscopy) and contractility (measured ventrally with traction microscopy), and increased mechanical anisotropy, without affecting either DNA synthesis or nuclear volume. Our results reveal that the nuclear volume in G1 is predictive of the proliferative status of single endothelial cells within a population, whereas cell stiffness and contractility are not. These findings show that the effects of cell mechanics in shape control of proliferation are far more complex than a linear or straightforward relationship. Our data are consistent with a mechanism by which spreading of cells in G1 partially enhances proliferation by inducing nuclear swelling and decreasing chromatin condensation, thereby rendering DNA more accessible to the replication machinery.
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Núcleo Celular/fisiología , Células Endoteliales/citología , Células Endoteliales/fisiología , Fase G1/fisiología , Mecanotransducción Celular/fisiología , Modelos Biológicos , Línea Celular , Proliferación Celular , Tamaño de la Célula , Simulación por Computador , Humanos , Estadística como AsuntoRESUMEN
The mechanical properties of the living cell are intimately related to cell signaling biology through cytoskeletal tension. The tension borne by the cytoskeleton (CSK) is in part generated internally by the actomyosin machinery and externally by stretch. Here we studied how cytoskeletal tension is modified during stretch and the tensional changes undergone by the sites of cell-matrix interaction. To this end we developed a novel technique to map cell-matrix stresses during application of stretch. We found that cell-matrix stresses increased with imposition of stretch but dropped below baseline levels on stretch release. Inhibition of the actomyosin machinery resulted in a larger relative increase in CSK tension with stretch and in a smaller drop in tension after stretch release. Cell-matrix stress maps showed that the loci of cell adhesion initially bearing greater stress also exhibited larger drops in traction forces after stretch removal. Our results suggest that stretch partially disrupts the actin-myosin apparatus and the cytoskeletal structures that support the largest CSK tension. These findings indicate that cells use the mechanical energy injected by stretch to rapidly reorganize their structure and redistribute tension.
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Citoesqueleto/fisiología , Células Epiteliales/fisiología , Matriz Extracelular/fisiología , Mecanotransducción Celular/fisiología , Modelos Biológicos , Línea Celular , Simulación por Computador , Elasticidad , Humanos , Estrés MecánicoRESUMEN
Collective cell migration is regulated by a complex set of mechanical interactions and cellular mechanisms. Collective migration emerges from mechanisms occurring at single cell level, involving processes like contraction, polymerization and depolymerization, of cell-cell interactions and of cell-substrate adhesion. Here, we present a computational framework which simulates the dynamics of this emergent behavior conditioned by substrates with stiffness gradients. The computational model reproduces the cell's ability to move toward the stiffer part of the substrate, process known as durotaxis. It combines the continuous formulation of truss elements and a particle-based approach to simulate the dynamics of cell-matrix adhesions and cell-cell interactions. Using this hybrid approach, researchers can quickly create a quantitative model to understand the regulatory role of different mechanical conditions on the dynamics of collective cell migration. Our model shows that durotaxis occurs due to the ability of cells to deform the substrate more in the part of lower stiffness than in the stiffer part. This effect explains why cell collective movement is more effective than single cell movement in stiffness gradient conditions. In addition, we numerically evaluate how gradient stiffness properties, cell monolayer size and force transmission between cells and extracellular matrix are crucial in regulating durotaxis.
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Movimiento Celular , Simulación por Computador , Algoritmos , Fenómenos Biomecánicos , Adhesión Celular , Proliferación Celular , Modelos BiológicosRESUMEN
The underlying mechanisms by which extracellular matrix (ECM) mechanics influences cell and tissue function remain to be elucidated because the events associated with this process span size scales from tissue to molecular level. Furthermore, ECM has an extremely complex hierarchical 3D structure and the load distribution is highly dependent on the architecture and mechanical properties of ECM. In the present study, the macro- and microscale mechanical properties of collagen gel were studied. Dynamic rheological testing was performed to study the macroscale mechanical properties of collagen gel. The microscale mechanical properties of collagen gel were measured using optical magnetic twisting cytometry (OMTC). Ferromagnetic beads embedded in the matrix were used as mechanical probes. Our study on the multiscale mechanical properties of collage matrix suggests several interesting differences between macro and microscale mechanical properties originated from the scales of measurements. At the macroscopic scale, storage and loss modulus increase with collagen concentrations. Nonaffine collagen fibril structural network deformation plays an important role in determining the macroscopic mechanical properties of the collagen matrix. At the microscopic scale, however, the local mechanical properties are less sensitive to changes in collagen concentration because of the more immediate/direct deformation of collagen fibrils in the OMTC measurements through forces exerted by locally attached ferromagnetic beads. The loss modulus is more affected by the local interstitial fluid environment, leading to a rather dramatic increase in viscosity with frequency, especially at higher frequencies (>10 Hz). A finite element model was developed to study the geometric factors in the OMTC measurements when the collagen matrix was considered to be hyperelastic. Our results show that the geometric factors are dependent on collagen concentration, or the stiffness of matrix, when nonlinear material properties of the matrix are considered, and thus interpretation of the apparent modulus from OMTC measurements should be conducted carefully.
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Contractile tension of alveolar epithelial cells plays a major role in the force balance that regulates the structural integrity of the alveolar barrier. The aim of this work was to study thrombin-induced contractile forces of alveolar epithelial cells. A549 alveolar epithelial cells were challenged with thrombin, and time course of contractile forces was measured by traction microscopy. The cells exhibited basal contraction with total force magnitude 55.0 +/- 12.0 nN (mean +/- SE, n = 12). Traction forces were exerted predominantly at the cell periphery and pointed to the cell center. Thrombin (1 U/ml) induced a fast and sustained 2.5-fold increase in traction forces, which maintained peripheral and centripetal distribution. Actin fluorescent staining revealed F-actin polymerization and enhancement of peripheral actin rim. Disruption of actin cytoskeleton with cytochalasin D (5 microM, 30 min) and inhibition of myosin light chain kinase with ML-7 (10 microM, 30 min) and Rho kinase with Y-27632 (10 microM, 30 min) markedly depressed basal contractile tone and abolished thrombin-induced cell contraction. Therefore, the contractile response of alveolar epithelial cells to the inflammatory agonist thrombin was mediated by actin cytoskeleton remodeling and actomyosin activation through myosin light chain kinase and Rho kinase signaling pathways. Thrombin-induced contractile tension might further impair alveolar epithelial barrier integrity in the injured lung.
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Adhesión Celular/fisiología , Alveolos Pulmonares/citología , Mucosa Respiratoria/citología , Mucosa Respiratoria/fisiología , Trombina/fisiología , Actinas/efectos de los fármacos , Actinas/fisiología , Actinas/ultraestructura , Amidas/farmacología , Azepinas/farmacología , Línea Celular , Citocalasina D/farmacología , Citoesqueleto/efectos de los fármacos , Citoesqueleto/fisiología , Citoesqueleto/ultraestructura , Humanos , Péptidos y Proteínas de Señalización Intracelular , Microscopía Fluorescente/métodos , Quinasa de Cadena Ligera de Miosina/antagonistas & inhibidores , Quinasa de Cadena Ligera de Miosina/fisiología , Naftalenos/farmacología , Permeabilidad , Proteínas Serina-Treonina Quinasas/antagonistas & inhibidores , Proteínas Serina-Treonina Quinasas/fisiología , Piridinas/farmacología , Transducción de Señal/fisiología , Factores de Tiempo , Quinasas Asociadas a rhoRESUMEN
The ability of cells to follow gradients of extracellular matrix stiffness-durotaxis-has been implicated in development, fibrosis, and cancer. Here, we found multicellular clusters that exhibited durotaxis even if isolated constituent cells did not. This emergent mode of directed collective cell migration applied to a variety of epithelial cell types, required the action of myosin motors, and originated from supracellular transmission of contractile physical forces. To explain the observed phenomenology, we developed a generalized clutch model in which local stick-slip dynamics of cell-matrix adhesions was integrated to the tissue level through cell-cell junctions. Collective durotaxis is far more efficient than single-cell durotaxis; it thus emerges as a robust mechanism to direct cell migration during development, wound healing, and collective cancer cell invasion.
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Células Epiteliales/fisiología , Matriz Extracelular , Taxia , Línea Celular , Humanos , Uniones Intercelulares/fisiología , Microscopía de Contraste de Fase , Miosinas/fisiologíaRESUMEN
Fundamental biological processes including morphogenesis and tissue repair require cells to migrate collectively. In these processes, epithelial or endothelial cells move in a cooperative manner coupled by intercellular junctions. Ultimately, the movement of these multicellular systems occurs through the generation of cellular forces, exerted either on the substrate via focal adhesions (cell-substrate forces) or on neighboring cells through cell-cell junctions (cell-cell forces). Quantitative measurements of multicellular forces and kinematics with cellular or subcellular resolution have become possible only in recent years. In this chapter, we describe some of these techniques, which include particle image velocimetry to map cell velocities, traction force microscopy to map forces exerted by cells on the substrate, and monolayer stress microscopy to map forces within and between cells. We also describe experimental protocols to perform these measurements. The combination of these techniques with high-resolution imaging tools and molecular perturbations will lead to a better understanding of the mechanisms underlying collective cell migration in health and disease.
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Movimiento Celular , Animales , Fenómenos Biomecánicos , Dimetilpolisiloxanos/química , Perros , Células de Riñón Canino Madin Darby , Fenómenos Magnéticos , Membranas Artificiales , RatasRESUMEN
For an organism to develop, for a wound to heal, or for a tumor to invade, cells must be able to migrate following directional cues. It is widely accepted that directed cell migration is enabled by cellular sensing of local gradients in the concentration of chemical factors. The main molecular players involved in this mode of cellular guidance--chemotaxis--have been identified and the combination of modeling and experimental approaches is progressively unveiling a clear picture of the underlying mechanisms. Evidence obtained over the past decade has shown that cells can also be guided by mechanical stimuli such as physical forces or gradients in extracellular matrix stiffness. Mechanical guidance, which we refer here globally as mechanotaxis, is also thought to drive processes in development, cancer, and wound healing, but experimental evidence is scattered and mechanisms remain largely unknown. Here we use the better understood process of chemotaxis as a reference to define the building blocks that are required for cell guidance, and then discuss how these building blocks might be organized in mechanotaxis. We show that both chemotaxis and mechanotaxis involve an exquisite interplay between physical and chemical mechanisms to sense gradients, establish polarization, and drive directed migration.
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Movimiento Celular , Quimiotaxis , Animales , Polaridad Celular , Matriz Extracelular , Humanos , Transducción de SeñalRESUMEN
Many fundamental cell processes, such as angiogenesis, neurogenesis and cancer metastasis, are thought to be modulated by extracellular matrix stiffness. Thus, the availability of matrix substrates having well-defined stiffness profiles can be of great importance in biophysical studies of cell-substrate interaction. Here, we present a method to fabricate biocompatible hydrogels with a well defined and linear stiffness gradient. This method, involving the photopolymerization of films by progressively uncovering an acrylamide/bis-acrylamide solution initially covered with an opaque mask, can be easily implemented with common lab equipment. It produces linear stiffness gradients of at least 115 kPa/mm, extending from â¼1 kPa to 240 kPa (in units of Young's modulus). Hydrogels with less steep gradients and narrower stiffness ranges can easily be produced. The hydrogels can be covalently functionalized with uniform coatings of proteins that promote cell adhesion. Cell spreading on these hydrogels linearly correlates with hydrogel stiffness, indicating that this technique effectively modifies the mechanical environment of living cells. This technique provides a simple approach that produces steeper gradients, wider rigidity ranges, and more accurate profiles than current methods.
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Movimiento Celular/fisiología , Matriz Extracelular/fisiología , Hidrogeles/metabolismo , Fenómenos Mecánicos , Acrilamida/química , Animales , Materiales Biocompatibles/química , Materiales Biocompatibles/metabolismo , Adhesión Celular/fisiología , Técnicas de Cultivo de Célula/métodos , Línea Celular Tumoral , Módulo de Elasticidad/efectos de la radiación , Matriz Extracelular/química , Fibronectinas/química , Fibronectinas/metabolismo , Humanos , Hidrogeles/química , Ratones , Microscopía de Fuerza Atómica , Células 3T3 NIH , Reproducibilidad de los Resultados , Soluciones/química , Rayos UltravioletaRESUMEN
Cell adhesion is required for essential biological functions such as migration, tissue formation and wound healing, and it is mediated by individual molecules that bind specifically to ligands on other cells or on the extracellular matrix. Atomic force microscopy (AFM) has been successfully used to measure cell adhesion at both single molecule and whole cell levels. However, the measurement of inherent cell adhesion properties requires a constant cell-probe contact area during indentation, a requirement which is not fulfilled in common pyramidal or spherical AFM tips. We developed a procedure using focused ion beam (FIB) technology by which we modified silicon pyramidal AFM cantilever tips to obtain flat-ended cylindrical tips with a constant and known area of contact. The tips were validated on elastic gels and living cells. Cylindrical tips showed a fairly linear force-indentation behaviour on both gels and cells for indentations >200 nm. Cylindrical tips coated with ligands were used to quantify inherent dynamic cell adhesion and elastic properties. Force, work of adhesion and elasticity showed a marked dynamic response. In contrast, the deformation applied to the cells before rupture was fairly constant within the probed dynamic range. Taken together, these results suggest that the dynamic adhesion strength is counterbalanced by the dynamic elastic response to keep a constant cell deformation regardless of the applied pulling rate.