Viscoelastic Properties of Confluent MDCK II Cells Obtained from Force Cycle Experiments

PyFMLab: Open-source software for atomic force microscopy microrheology data analysis

Background

Atomic force microscopy (AFM) is one of the main techniques used to characterize the mechanical properties of soft biological samples and biomaterials at the nanoscale. Despite efforts made by the AFM community to promote open-source data analysis tools, standardization continues to be a significant concern in a field that requires common analysis procedures. AFM-based mechanical measurements involve applying a controlled force to the sample and measure the resulting deformation in the so-called force-distance curves. These may include simple approach and retract or oscillatory cycles at various frequencies (microrheology). To extract quantitative parameters, such as the elastic modulus, from these measurements, AFM measurements are processed using data analysis software. Although open tools exist and allow obtaining the mechanical properties of the sample, most of them only include standard elastic models and do not allow the processing of microrheology data. In this work, we have developed an open-source software package (called PyFMLab, as of python force microscopy laboratory) capable of determining the viscoelastic properties of samples from both conventional force-distance curves and microrheology measurements.

Methods

PyFMLab has been written in Python, which provides an accessible syntax and sufficient computational efficiency. The software features were divided into separate, self-contained libraries to enhance code organization and modularity and to improve readability, maintainability, testability, and reusability. To validate PyFMLab, two AFM datasets, one composed of simple force curves and another including oscillatory measurements, were collected on HeLa cells.

Results

The viscoelastic parameters obtained on the two datasets analysed using PyFMLab were validated against data processing proprietary software and against validated MATLAB routines developed before obtaining equivalent results.

Conclusions

Its open-source nature and versatility makes PyFMLab an open-source solution that paves the way for standardized viscoelastic characterization of biological samples from both force-distance curves and microrheology measurements.

Measuring the viscoelastic relaxation function of cells with a time-dependent interpretation of the Hertz-Sneddon indentation model

The Hertz-Sneddon elastic indentation model is widely adopted in the biomechanical investigation of living cells and other soft materials using atomic force microscopy despite the explicit viscoelastic nature of these materials. In this work, we demonstrate that an exact analytical viscoelastic force model for power-law materials, can be interpreted as a time-dependent Hertz-Sneddon-like model. Characterizing fibroblasts (L929) and osteoblasts (OFCOLII) demonstrates the model's accuracy. Our results show that the difference between Young's modulus E Y obtained by fitting force curves with the Hertz-Sneddon model and the effective Young's modulus derived from the viscoelastic force model is less than 3%, even when cells are probed at large forces where nonlinear deformation effects become significant. We also propose a measurement protocol that involves probing samples at different indentation speeds and forces, enabling the construction of the average viscoelastic relaxation function of samples by conveniently fitting the force curves with the Hertz-Sneddon model.

Nanorheology and Nanoindentation Revealed a Softening and an Increased Viscous Fluidity of Adherent Mammalian Cells upon Increasing the Frequency

The nanomechanical response of a cell depends on the frequency at which the cell is probed. The components of the cell that contribute to this property and their interplay are not well understood. Here, two force microscopy methods are integrated to characterize the frequency and/or the velocity-dependent properties of living cells. It is shown on HeLa and fibroblasts, that cells soften and fluidize upon increasing the frequency or the velocity of the deformation. This property was independent of the type and values (25 or 1000 nm) of the deformation. At low frequencies (2-10 Hz) or velocities (1-10 µm s-1 ), the response is dominated by the mechanical properties of the cell surface. At higher frequencies (>10 Hz) or velocities (>10 µm s-1 ), the response is dominated by the hydrodynamic drag of the cytosol. Softening and fluidization does not seem to involve any structural remodeling. It reflects a redistribution of the applied stress between the solid and liquid-like elements of the cell as the frequency or the velocity is changed. The data indicates that the quasistatic mechanical properties of a cell featuring a cytoskeleton pathology might be mimicked by the response of a non-pathological cell which is probed at a high frequency.

Coupled mechanical mapping and interference contrast microscopy reveal viscoelastic and adhesion hallmarks of monocyte differentiation into macrophages

Monocytes activated by pro-inflammatory signals adhere to the vascular endothelium and migrate from the bloodstream to the tissue ultimately differentiating into macrophages. Cell mechanics and adhesion play a crucial role in macrophage functions during this inflammatory process. However, how monocytes change their adhesion and mechanical properties upon differentiation into macrophages is still not well understood. In this work, we used various tools to quantify the morphology, adhesion, and viscoelasticity of monocytes and differentiatted macrophages. Combination of atomic force microscopy (AFM) high resolution viscoelastic mapping with interference contrast microscopy (ICM) at the single-cell level revealed viscoelasticity and adhesion hallmarks during monocyte differentiation into macrophages. Quantitative holographic tomography imaging revealed a dramatic increase in cell volume and surface area during monocyte differentiation and the emergence of round and spread macrophage subpopulations. AFM viscoelastic mapping showed important stiffening (increase of the apparent Young's modulus, E₀) and solidification (decrease of cell fluidity, β) on differentiated cells that correlated with increased adhesion area. These changes were enhanced in macrophages with a spread phenotype. Remarkably, when adhesion was perturbed, differentiated macrophages remained stiffer and more solid-like than monocytes, suggesting a permanent reorganization of the cytoskeleton. We speculate that the stiffer and more solid-like microvilli and lamellipodia might help macrophages to minimize energy dissipation during mechanosensitive activities. Thus, our results revealed viscoelastic and adhesion hallmarks of monocyte differentiation that may be important for biological function.

Optimizing the accuracy of viscoelastic characterization with AFM force-distance experiments in the time and frequency domains

Atomic Force Microscopy (AFM) force-distance (FD) experiments have emerged as an attractive alternative to traditional micro-rheology measurement techniques owing to their versatility of use in materials of a wide range of mechanical properties. Here, we show that the range of time dependent behaviour which can reliably be resolved from the typical method of FD inversion (fitting constitutive FD relations to FD data) is inherently restricted by the experimental parameters: sampling frequency, experiment length, and strain rate. Specifically, we demonstrate that violating these restrictions can result in errors in the values of the parameters of the complex modulus. In the case of complex materials, such as cells, whose behaviour is not specifically understood a priori, the physical sensibility of these parameters cannot be assessed and may lead to falsely attributing a physical phenomenon to an artifact of the violation of these restrictions. We use arguments from information theory to understand the nature of these inconsistencies as well as devise limits on the range of mechanical parameters which can be reliably obtained from FD experiments. The results further demonstrate that the nature of these restrictions depends on the domain (time or frequency) used in the inversion process, with the time domain being far more restrictive than the frequency domain. Finally, we demonstrate how to use these restrictions to better design FD experiments to target specific timescales of a material's behaviour through our analysis of a polydimethylsiloxane (PDMS) polymer sample.

Effects of vimentin on the migration, search efficiency, and mechanical resilience of dendritic cells

Dendritic cells use amoeboid migration to pass through narrow passages in the extracellular matrix and confined tissue in search for pathogens and to reach the lymph nodes and alert the immune system. Amoeboid migration is a migration mode that, instead of relying on cell adhesion, is based on mechanical resilience and friction. To better understand the role of intermediate filaments in ameboid migration, we studied the effects of vimentin on the migration of dendritic cells. We show that the lymph node homing of vimentin-deficient cells is reduced in our in vivo experiments in mice. Lack of vimentin also reduces the cell stiffness, the number of migrating cells, and the migration speed in vitro in both 1D and 2D confined environments. Moreover, we find that lack of vimentin weakens the correlation between directional persistence and migration speed. Thus, vimentin-expressing dendritic cells move faster in straighter lines. Our numerical simulations of persistent random search in confined geometries verify that the reduced migration speed and the weaker correlation between the speed and direction of motion result in longer search times to find regularly located targets. Together, these observations show that vimentin enhances the ameboid migration of dendritic cells, which is relevant for the efficiency of their random search for pathogens.

Epithelial cells sacrifice excess area to preserve fluidity in response to external mechanical stress

Viscoelastic properties of epithelial cells subject to shape changes were monitored by indentation-retraction/relaxation experiments. MDCK II cells cultured on extensible polydimethylsiloxane substrates were laterally stretched and, in response, displayed increased cortex contractility and loss of excess surface area. Thereby, the cells preserve their fluidity but inevitably become stiffer. We found similar behavior in demixed cell monolayers of ZO-1/2 double knock down (dKD) cells, cells exposed to different temperatures and after removal of cholesterol from the plasma membrane. Conversely, the mechanical response of single cells adhered onto differently sized patches displays no visible rheological change. Sacrificing excess surface area allows the cells to respond to mechanical challenges without losing their ability to flow. They gain a new degree of freedom that permits resolving the interdependence of fluidity β on stiffness [Formula: see text]. We also propose a model that permits to tell apart contributions from excess membrane area and excess cell surface area.

Foregut organ progenitors and their niche display distinct viscoelastic properties in vivo during early morphogenesis stages

Material properties of living matter play an important role for biological function and development. Yet, quantification of material properties of internal organs in vivo, without causing physiological damage, remains challenging. Here, we present a non-invasive approach based on modified optical tweezers for quantifying sub-cellular material properties deep inside living zebrafish embryos. Material properties of cells within the foregut region are quantified as deep as 150 µm into the biological tissue through measurements of the positions of an inert tracer. This yields an exponent, α, which characterizes the scaling behavior of the positional power spectra and the complex shear moduli. The measurements demonstrate differential mechanical properties: at the time when the developing organs undergo substantial displacements during morphogenesis, gut progenitors are more elastic (α = 0.57 ± 0.07) than the neighboring yolk (α = 0.73 ± 0.08), liver (α = 0.66 ± 0.06) and two mesodermal (α = 0.68 ± 0.06, α = 0.64 ± 0.06) progenitor cell populations. The higher elasticity of gut progenitors correlates with an increased cellular concentration of microtubules. The results infer a role of material properties during morphogenesis and the approach paves the way for quantitative material investigations in vivo of embryos, explants, or organoids.

Here, the authors present a method based on optical tweezers to measure mechanical properties of cells inside living zebrafish embryos. The measurement reveals spatiotemporally distinct mechanical properties, linking cell mechanics and morphogenesis.

B. F. Lima MA, Zanetti M, Rubert A, Ciubotaru C, Lazzarino M, Sbaizero O, Cojoc D. The Role of Cytoskeleton Revealed by Quartz Crystal Microbalance and Digital Holographic Microscopy

The connection between cytoskeleton alterations and diseases is well known and has stimulated research on cell mechanics, aiming to develop reliable biomarkers. In this study, we present results on rheological, adhesion, and morphological properties of primary rat cardiac fibroblasts, the cytoskeleton of which was altered by treatment with cytochalasin D (Cyt-D) and nocodazole (Noc), respectively. We used two complementary techniques: quartz crystal microbalance (QCM) and digital holographic microscopy (DHM). Qualitative data on cell viscoelasticity and adhesion changes at the cell–substrate near-interface layer were obtained with QCM, while DHM allowed the measurement of morphological changes due to the cytoskeletal alterations. A rapid effect of Cyt-D was observed, leading to a reduction in cell viscosity, loss of adhesion, and cell rounding, often followed by detachment from the surface. Noc treatment, instead, induced slower but continuous variations in the rheological behavior for four hours of treatment. The higher vibrational energy dissipation reflected the cell’s ability to maintain a stable attachment to the substrate, while a cytoskeletal rearrangement occurs. In fact, along with the complete disaggregation of microtubules at prolonged drug exposure, a compensatory effect of actin polymerization emerged, with increased stress fiber formation.

An explicit model to extract viscoelastic properties of cells from AFM force-indentation curves

Atomic force microscopy (AFM) is widely used for quantifying the mechanical properties of soft materials such as cells. AFM force-indentation curves are conventionally fitted with a Hertzian model to extract elastic properties. These properties solely are, however, insufficient to describe the mechanical properties of cells. Here, we expand the analysis capabilities to describe the viscoelastic behavior while using the same force-indentation curves. Our model gives an explicit relation of force and indentation and extracts physically meaningful mechanical parameters. We first validated the model on simulated force-indentation curves. Then, we applied the fitting model to the force-indentation curves of two hydrogels with different crosslinking mechanisms. Finally, we characterized HeLa cells in two cell cycle phases, interphase and mitosis, and showed that mitotic cells have a higher apparent elasticity and a lower apparent viscosity. Our study provides a simple method, which can be directly integrated into the standard AFM framework for extracting the viscoelastic properties of materials.

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Simple mechanical model to describe viscoelastic properties of soft matter

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A model fitted directly to force-indentation curves

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Capturing the distinct nature of hydrogels crosslinked in different mechanisms

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Comparing viscoelastic properties of cells in interphase and mitotic states

Epithelial cells fluidize upon adhesion but display mechanical homeostasis in the adherent state

Atomic force microscopy is used to study the viscoelastic properties of epithelial cells in three different states. Force relaxation data are acquired from cells in suspension, adhered but single cells, and polarized cells in a confluent monolayer using different indenter geometries comprising flat bars, pyramidal cones, and spheres. We found that the fluidity of cells increased substantially from the suspended to the adherent state. Along this line, the prestress of suspended cells generated by cortical contractility is also greater than that of cells adhering to a surface. Polarized cells that are part of a confluent monolayer form an apical cap that is soft and fluid enough to respond rapidly to mechanical challenges from wounding, changes in the extracellular matrix, osmotic stress, and external deformation. In contrast to adherent cells, cells in the suspended state show a pronounced dependence of fluidity on the external areal strain. With increasing areal strain, the suspended cells become softer and more fluid. We interpret the results in terms of cytoskeletal remodeling that softens cells in the adherent state to facilitate adhesion and spreading by relieving internal active stress. However, once the cells spread on the surface they maintain their mechanical phenotype displaying viscoelastic homeostasis.

Nanophysical Mapping of Inflammasome Activation by Nanoparticles via Specific Cell Surface Recognition Events

Silica nanoparticles (SiNP) trigger a range of innate immune responses in relevant essential organs, such as the liver and the lungs. Inflammatory reactions, including NLRP3 inflammasome activation, have been linked to particulate materials; however, the molecular mechanisms and key actors remain elusive. Although many receptors, including several scavenger receptors, were suggested to participate in SiNP cellular uptake, mechanistic evidence of their role on innate immunity is lacking. Here we present an atomic force microscopy-based approach to physico-mechanically map the specific interaction occurring between nanoparticles and scavenger receptor A1 (SRA1) in vitro on living lung epithelial cells. We find that SiNP recognition by SRA1 on human macrophages plays a key role in mediating NLRP3 inflammasome activation, and we identify cellular mechanical changes as clear indicators of inflammasome activation in human macrophages, greatly advancing our knowledge on the interplay among nanomaterials and innate immunity.

Viscoelastic properties of epithelial cells

Epithelial cells form tight barriers that line both the outer and inner surfaces of organs and cavities and therefore face diverse environmental challenges. The response to these challenges relies on the cells' dynamic viscoelastic properties, playing a pivotal role in many biological processes such as adhesion, growth, differentiation, and motility. Therefore, the cells usually adapt their viscoelastic properties to mirror the environment that determines their fate and vitality. Albeit not a high-throughput method, atomic force microscopy is still among the dominating methods to study the mechanical properties of adherent cells since it offers a broad range of forces from Piconewtons to Micronewtons at biologically significant time scales. Here, some recent work of deformation studies on epithelial cells is reviewed with a focus on viscoelastic models suitable to describe force cycle measurements congruent with the architecture of the actin cytoskeleton. The prominent role of the cortex in the cell's response to external forces is discussed also in the context of isolated cortex extracts on porous surfaces.

Viscoelasticity of basal plasma membranes and cortices derived from MDCK II cells

The mechanical properties of cells are largely determined by the architecture and dynamics of their viscoelastic cortex, which consists of a contractile, cross-linked actin mesh attached to the plasma membrane via linker proteins. Measuring the mechanical properties of adherent, polarized epithelial cells is usually limited to the upper, i.e., apical side, of the cells because of their accessibility on culture dishes. Therefore, less is known about the viscoelastic properties of basal membranes. Here, I investigate the viscoelastic properties of basolateral membranes derived from polarized MDCK II epithelia in response to external deformation and compare them to living cells probed at the apical side. MDCK II cells were grown on porous surfaces to confluence, and the upper cell body was removed via a squirting-lysis protocol. The free-standing, defoliated basal membranes were subject to force indentation and relaxation experiments permitting a precise assessment of cortical viscoelasticity. A new theoretical framework to describe the force cycles is developed and applied to obtain the time-dependent area compressibility modulus of cell cortices from adherent cells. Compared with the viscoelastic response of living cells, the basolateral membranes are substantially less fluid and stiffer but obey to the same universal scaling law if excess area is taken correctly into account.

The viscoelasticity of adherent cells follows a single power-law with distinct local variations within a single cell and across cell lines

AFM-based force-distance curves are commonly used to characterize the nanomechanical properties of live cells. The transformation of these curves into nanomechanical properties requires the development of contact mechanics models. Spatially-resolved force-distance curves involving 1 to 2 μm deformations were obtained on HeLa and NIH 3T3 (fibroblast) cells. An elastic and two viscoelastic models were used to describe the experimental force-distance curves. The best agreement was obtained by applying a contact mechanics model that accounts for the geometry of the contact and the finite-thickness of the cell and assumes a single power-law dependence with time. Our findings show the shortcomings of elastic and semi-infinite viscoelastic models to characterize the mechanical response of a mammalian cell under micrometer-scale deformations. The parameters of the 3D power-law viscoelastic model, compressive modulus and fluidity exponent showed local variations within a single cell and across the two cell lines. The corresponding nanomechanical maps revealed structures that were not visible in the AFM topographic maps.

Viscoelasticity and Noise Properties Reveal the Formation of Biomemory in Cells

Living cells are neither perfectly elastic nor liquid and return a viscoelastic response to external stimuli. Nanoindentation provides force–distance curves, allowing the investigation of cell mechanical properties, and yet, these curves can differ from point to point on the cell surface, revealing its inhomogeneous character. In the present work, we propose a mathematical method to estimate both viscoelastic and noise properties of cells as these are depicted on the values of the scaling exponents of relaxation function and power spectral density, respectively. The method uses as input the time derivative of the response force in a nanoindentation experiment. Generalized moments method and/or rescaled range analysis is used to study the resulting time series depending on their nonstationary or stationary nature. We conducted experiments in living Ulocladium chartarum spores. We found that spores in the approaching phase present a viscoelastic behavior with the corresponding scaling exponent in the range 0.25–0.52 and in the retracting phase present a liquid-like behavior with exponents in the range 0.67–0.85. This substantial difference of the scaling exponents in the two phases suggests the formation of biomemory as a response of the spores to the indenting AFM mechanical stimulus. The retracting phase may be described as a process driven by bluish noises, while the approaching one is driven by persistent noise.

The nature of cell division forces in epithelial monolayers

Epithelial cells undergo striking morphological changes during division to ensure proper segregation of genetic and cytoplasmic materials. These morphological changes occur despite dividing cells being mechanically restricted by neighboring cells, indicating the need for extracellular force generation. Beyond driving cell division itself, forces associated with division have been implicated in tissue-scale processes, including development, tissue growth, migration, and epidermal stratification. While forces generated by mitotic rounding are well understood, forces generated after rounding remain unknown. Here, we identify two distinct stages of division force generation that follow rounding: (1) Protrusive forces along the division axis that drive division elongation, and (2) outward forces that facilitate postdivision spreading. Cytokinetic ring contraction of the dividing cell, but not activity of neighboring cells, generates extracellular forces that propel division elongation and contribute to chromosome segregation. Forces from division elongation are observed in epithelia across many model organisms. Thus, division elongation forces represent a universal mechanism that powers cell division in confining epithelia.

Multi-oscillation microrheology via acoustic force spectroscopy enables frequency-dependent measurements on endothelial cells at high-throughput

Active microrheology is one of the main methods to determine the mechanical properties of cells and tissue, and the modelling of these viscoelastic properties is under heavy debate with many competing approaches. Most experimental methods of active microrheology such as optical tweezers or atomic force microscopy based approaches rely on single cell measurements, and thus suffer from a low throughput. Here, we present a novel method for frequency-dependent microrheology on cells using acoustic forces which allows multiplexed measurements of several cells in parallel. Acoustic force spectroscopy (AFS) is used to generate multi-oscillatory forces in the range of pN-nN on particles attached to primary human umbilical vein endothelial cells (HUVEC) cultivated inside a microfluidic chip. While the AFS was introduced as a single-molecule technique to measure mechanochemical properties of biomolecules, we exploit the AFS to measure the dynamic viscoelastic properties of cells exposed to different conditions, such as flow shear stresses or drug injections. By controlling the force and measuring the position of the particle, the complex shear modulus G*(ω) can be measured continuously over several hours. The resulting power-law shear moduli are consistent with fractional viscoelastic models. In our experiments we confirm a decrease in shear modulus after perturbing the actin cytoskeleton via cytochalasin B. This effect was reversible after washing out the drug. Additionally, we include critical information for the usage of the new method AFS as a measurement tool showing its capabilities and limitations and we find that for performing viscoelastic measurements with the AFS, a thorough calibration and careful data analysis is crucial, for which we provide protocols and guidelines.

Human mammary epithelial cells in a mature, stratified epithelial layer flatten and stiffen compared to single and confluent cells

BACKGROUND

The epithelium forms a protective barrier against external biological, chemical and physical insults. So far, AFM-based, micro-mechanical measurements have only been performed on single cells and confluent cells, but not yet on cells in mature layers.

METHODS

Using a combination of atomic force, fluorescence and confocal microscopy, we determined the changes in stiffness, morphology and actin distribution of human mammary epithelial cells (HMECs) as they transition from single cells to confluency to a mature layer.

RESULTS

Single HMECs have a tall, round (planoconvex) morphology, have actin stress fibers at the base, have diffuse cortical actin, and have a stiffness of 1 kPa. Confluent HMECs start to become flatter, basal actin stress fibers start to disappear, and actin accumulates laterally where cells abut. Overall stiffness is still 1 kPa with two-fold higher stiffness in the abutting regions. As HMECs mature and form multilayered structures, cells on apical surfaces become flatter (apically more level), wider, and seven times stiffer (mean, 7 kPa) than single and confluent cells. The main drivers of these changes are actin filaments, as cells show strong actin accumulation in the regions where cells adjoin, and in the apical regions.

CONCLUSIONS

HMECs stiffen, flatten and redistribute actin upon transiting from single cells to mature, confluent layers.

GENERAL SIGNIFICANCE

Our findings advance the understanding of breast ductal morphogenesis and mechanical homeostasis.

Mechanical Characterization for Cellular Mechanobiology: Current Trends and Future Prospects

Accurate mechanical characterization of adherent cells and their substrates is important for understanding the influence of mechanical properties on cells themselves. Recent mechanobiology studies outline the importance of mechanical parameters, such as stress relaxation and strain stiffening on the behavior of cells. Numerous techniques exist for probing mechanical properties and it is vital to understand the benefits of each technique and how they relate to each other. This mini review aims to guide the reader through the toolbox of mechanical characterization techniques by presenting well-established and emerging methods currently used to assess mechanical properties of substrates and cells.

Viscoelasticity of Native and Artificial Actin Cortices Assessed by Nanoindentation Experiments

Cell cortices are responsible for the resilience and morphological dynamics of cells. Measuring their mechanical properties is impeded by contributions from other filament types, organelles, and the crowded cytoplasm. We established a versatile concept for the precise assessment of cortical viscoelasticity based on force cycle experiments paired with continuum mechanics. Apical cell membranes of confluent MDCK II cells were deposited on porous substrates and locally deformed. Force cycles could be described with a time-dependent area compressibility modulus obeying the same power law as employed for whole cells. The reduced fluidity of apical cell membranes compared to living cells could partially be restored by reactivating myosin motors. A comparison with artificial minimal actin cortices (MACs) reveals lower stiffness and higher fluidity attributed to missing cross-links in MACs.

Buckling of an Epithelium Growing under Spherical Confinement

Many organs are formed through folding of an epithelium. This change in shape is usually attributed to tissue heterogeneities, for example, local apical contraction. In contrast, compressive stresses have been proposed to fold a homogeneous epithelium by buckling. While buckling is an appealing mechanism, demonstrating that it underlies folding requires measurement of the stress field and the material properties of the tissue, which are currently inaccessible in vivo. Here, we show that monolayers of identical cells proliferating on the inner surface of elastic spherical shells can spontaneously fold. By measuring the elastic deformation of the shell, we infer the forces acting within the monolayer and its elastic modulus. Using analytical and numerical theories linking forces to shape, we find that buckling quantitatively accounts for the shape changes of our monolayers. Our study shows that forces arising from epithelial growth in three-dimensional confinement are sufficient to drive folding by buckling.

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A proliferating epithelium encapsulated in a hollow sphere spontaneously invaginates

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Epithelial proliferation generates compressive stresses that deform the elastic shell

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Coupling between stress and folding shape shows that folding arises from buckling

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Epithelial elastic moduli are inferred from buckling theory and experiments

Prestress and Area Compressibility of Actin Cortices Determine the Viscoelastic Response of Living Cells

Shape, dynamics, and viscoelastic properties of eukaryotic cells are primarily governed by a thin, reversibly cross-linked actomyosin cortex located directly beneath the plasma membrane. We obtain time-dependent rheological responses of fibroblasts and MDCK II cells from deformation-relaxation curves using an atomic force microscope to access the dependence of cortex fluidity on prestress. We introduce a viscoelastic model that treats the cell as a composite shell and assumes that relaxation of the cortex follows a power law giving access to cortical prestress, area-compressibility modulus, and the power law exponent (fluidity). Cortex fluidity is modulated by interfering with myosin activity. We find that the power law exponent of the cell cortex decreases with increasing intrinsic prestress and area-compressibility modulus, in accordance with previous finding for isolated actin networks subject to external stress. Extrapolation to zero tension returns the theoretically predicted power law exponent for transiently cross-linked polymer networks. In contrast to the widely used Hertzian mechanics, our model provides viscoelastic parameters independent of indenter geometry and compression velocity.

Viscoelasticity in simple indentation-cycle experiments: a computational study

Instrumented indentation has become an indispensable tool for quantitative analysis of the mechanical properties of soft polymers and biological samples at different length scales. These types of samples are known for their prominent viscoelastic behavior, and attempts to calculate such properties from the indentation data are constantly made. The simplest indentation experiment presents a cycle of approach (deepening into the sample) and retraction of the indenter, with the output of the force and indentation depth as functions of time and a force versus indentation dependency (force curve). The linear viscoelastic theory based on the elastic–viscoelastic correspondence principle might predict the shape of force curves based on the experimental conditions and underlying relaxation function of the sample. Here, we conducted a computational analysis based on this theory and studied how the force curves were affected by the indenter geometry, type of indentation (triangular or sinusoidal ramp), and the relaxation functions. The relaxation functions of both traditional and fractional viscoelastic models were considered. The curves obtained from the analytical solutions, numerical algorithm and finite element simulations matched each other well. Common trends for the curve-related parameters (apparent Young’s modulus, normalized hysteresis area, and curve exponent) were revealed. Importantly, the apparent Young’s modulus, obtained by fitting the approach curve to the elastic model, demonstrated a direct relation to the relaxation function for all the tested cases. The study will help researchers to verify which model is more appropriate for the sample description without extensive calculations from the basic curve parameters and their dependency on the indentation rate.

Nanomechanical mapping of soft materials with the atomic force microscope: methods, theory and applications

Fast, high-resolution, non-destructive and quantitative characterization methods are needed to develop materials with tailored properties at the nanoscale or to understand the relationship between mechanical properties and cell physiology. This review introduces the state-of-the-art force microscope-based methods to map at high-spatial resolution the elastic and viscoelastic properties of soft materials. The experimental methods are explained in terms of the theories that enable the transformation of observables into material properties. Several applications in materials science, molecular biology and mechanobiology illustrate the scope, impact and potential of nanomechanical mapping methods.

Nanorheology of living cells measured by AFM-based force-distance curves

Mechanobiology aims to establish functional relationships between the mechanical state of a living a cell and its physiology. The acquisition of force-distance curves with an AFM is by far the dominant method to characterize the nanomechanical properties of living cells. However, theoretical simulations have shown that the contact mechanics models used to determine the Young's modulus from a force-distance curve could be off by a factor 5 from its expected value. The semi-quantitative character arises from the lack of a theory that integrates the AFM data, a realistic viscoelastic model of a cell and its finite-thickness. Here, we develop a method to determine the mechanical response of a cell from a force-distance curve. The method incorporates bottom-effect corrections, a power-law rheology model and the deformation history of the cell. It transforms the experimental data into viscoelastic parameters of the cell as a function of the indentation frequency. The quantitative agreement obtained between the experiments performed on living fibroblast cells and the analytical theory supports the use of force-distance curves to measure the nanorheological properties of cells.

Measuring viscoelasticity of soft biological samples using atomic force microscopy

Mechanical properties play important roles at different scales in biology. At the level of a single cell, the mechanical properties mediate mechanosensing and mechanotransduction, while at the tissue and organ levels, changes in mechanical properties are closely connected to disease and physiological processes. Over the past three decades, atomic force microscopy (AFM) has become one of the most widely used tools in the mechanical characterization of soft samples, ranging from molecules, cell organoids and cells to whole tissue. AFM methods can be used to quantify both elastic and viscoelastic properties, and significant recent developments in the latter have been enabled by the introduction of new techniques and models for data analysis. Here, we review AFM techniques developed in recent years for examining the viscoelastic properties of cells and soft gels, describe the main steps in typical data acquisition and analysis protocols, and discuss relevant viscoelastic models and how these have been used to characterize the specific features of cellular and other biological samples. We also discuss recent trends and potential directions for this field.

Viscoelastic mapping of cells based on fast force volume and PeakForce Tapping

Development of fast force volume (FFV), PeakForce Tapping (PFT), and related AFM techniques allow fast acquisition and mapping of a sample's mechanical properties. The methods are well-suited for studying soft biological samples like living cells in a liquid environment. However, the question remains how the measured mechanical properties are related to those acquired with the classical force volume (FV) technique conducted at low indentation rates. The difference is coming mostly from the pronounced viscoelastic behavior of cells, making apparent elastic parameters depending on the probing rate. Here, the viscoelastic analysis was applied directly to the force curves acquired with force volume or PeakForce Tapping by their post-processing based on the Ting's model. Maps from classical force volume, FFV and PFT obtained using special PFT cantilevers and cantilevers modified with microspheres were compared here. With the correct viscoelastic model, which was found to be the power-law rheology model, all the techniques have provided self-consistent results. The techniques were further modified for the mapping of the viscoelastic model-independent complex Young's modulus.

Integrated elastomer-based device for measuring the mechanics of adherent cell monolayers

Cells in the body collectively sustain mechanical deformations in almost all physiological functions. From the morphogenesis stage, cells' ability to sustain stress is essential for the body's well-being. Several pathologies have been associated with abnormal mechanical properties, thus suggesting the Young's modulus as a biomarker to diagnose diseases and determine their progression. Advancements in the field are quite slow because current techniques for measuring cell and tissue mechanics rely on complex and bulky measurement platforms that have low repeatability rates and limited measurement time-scales. We present the first miniaturized system that allows accurate quantification of the Young's modulus of adherent cell monolayers over a longer time (1-2 days). Our approach is based on tensile testing and optical read-out. Thanks to a thoughtful design and material choice, we are able to miniaturize tensile testing platforms into a 1 cm × 2 cm device. We provide highly repeatable Young's modulus measurements in the relevant range between 3 kPa and 300 kPa, over time and under physiological conditions, thus representing an interesting alternative to existing measurement platforms. Furthermore, the compatibility with standard biological equipment, continuous optical imaging and measurements on all types of adherent cells make this device highly versatile. Measurements on human sarcoma osteogenic (SaOS2) and Madin-Darby canine kidney cells (MDCK) are reported. The demonstrated capability to measure real-time changes in mechanical properties, such as after chemical treatment, opens the door for investigating the effects of drugs on cell mechanics.

Stiffness of MDCK II Cells Depends on Confluency and Cell Size

Mechanical phenotyping of adherent cells has become a serious tool in cell biology to understand how cells respond to their environment and eventually to identify disease patterns such as the malignancy of cancer cells. In the steady state, homeostasis is of pivotal importance, and cells strive to maintain their internal stresses even in challenging environments and in response to external chemical and mechanical stimuli. However, a major problem exists in determining mechanical properties because many techniques, such as atomic force microscopy, that assess these properties of adherent cells locally can only address a limited number of cells and provide elastic moduli that vary substantially from cell to cell. The origin of this spread in stiffness values is largely unknown and might limit the significance of measurements. Possible reasons for the disparity are variations in cell shape and size, as well as biological reasons such as the cell cycle or polarization state of the cell. Here, we show that stiffness of adherent epithelial cells rises with increasing projected apical cell area in a nonlinear fashion. This size stiffening not only occurs as a consequence of varying cell-seeding densities, it can also be observed within a small area of a particular cell culture. Experiments with single adherent cells attached to defined areas via microcontact printing show that size stiffening is limited to cells of a confluent monolayer. This leads to the conclusion that cells possibly regulate their size distribution through cortical stress, which is enhanced in larger cells and reduced in smaller cells.

Anisotropy vs isotropy in living cell indentation with AFM

The measurement of local mechanical properties of living cells by nano/micro indentation relies on the foundational assumption of locally isotropic cellular deformation. As a consequence of assumed isotropy, the cell membrane and underlying cytoskeleton are expected to locally deform axisymmetrically when indented by a spherical tip. Here, we directly observe the local geometry of deformation of membrane and cytoskeleton of different living adherent cells during nanoindentation with the integrated Atomic Force (AFM) and spinning disk confocal (SDC) microscope. We show that the presence of the perinuclear actin cap (apical stress fibers), such as those encountered in cells subject to physiological forces, causes a strongly non-axisymmetric membrane deformation during indentation reflecting local mechanical anisotropy. In contrast, axisymmetric membrane deformation reflecting mechanical isotropy was found in cells without actin cap: cancerous cells MDA-MB-231, which naturally lack the actin cap, and NIH 3T3 cells in which the actin cap is disrupted by latrunculin A. Careful studies were undertaken to quantify the effect of the live cell fluorescent stains on the measured mechanical properties. Using finite element computations and the numerical analysis, we explored the capability of one of the simplest anisotropic models – transverse isotropy model with three local mechanical parameters (longitudinal and transverse modulus and planar shear modulus) – to capture the observed non-axisymmetric deformation. These results help identifying which cell types are likely to exhibit non-isotropic properties, how to measure and quantify cellular deformation during AFM indentation using live cell stains and SDC, and suggest modelling guidelines to recover quantitative estimates of the mechanical properties of living cells.

Determination of the viscoelastic properties of a single cell cultured on a rigid support by force microscopy

Understanding the relationship between the mechanical properties of living cells and physiology is a central issue in mechanobiology. Mechanical properties are used as fingerprints of the pathological state of a single cell. The force exerted on a cell is influenced by the stiffness of the solid support needed to culture it. This effect is a consequence of the cell's boundary conditions. It causes a cell to appear with mechanical properties different from their real values. Here we develop a bottom effect viscoelastic theory to determine the viscoelastic response of a cell. The theory transforms a force-distance curve into the cell's Young's modulus, loss modulus, relaxation time or viscosity coefficient with independence of the stiffness of the rigid support. The theory predicts that, for a given indentation, the force exerted on the cell's periphery will be larger than on a perinuclear region. Results based on the use of semi-infinite contact mechanics models introduce large numerical errors in the determination of the mechanical properties. Finite element simulations confirm the theory and define its range of applicability.

Viscoelastic properties of human pancreatic tumors and in vitro constructs to mimic mechanical properties

Pancreatic ductal adenocarcinoma (PDAC) is almost universally fatal, in large part due to a protective fibrotic barrier generated by tumor-associated stromal (TAS) cells. This barrier is thought to promote cancer cell survival and confounds attempts to develop effective therapies. We present a 3D in vitro system that replicates the mechanical properties of the PDAC microenvironment, representing an invaluable tool for understanding the biology of the disease. Mesoscale indentation quantified viscoelastic metrics of resected malignant tumors, inflamed chronic pancreatitis regions, and histologically normal tissue. Both pancreatitis (2.15 ± 0.41 kPa, Mean ± SD) and tumors (5.46 ± 3.18 kPa) exhibit higher Steady-State Modulus (SSM) than normal tissue (1.06 ± 0.25 kPa; p < .005). The average viscosity of pancreatitis samples (63.2 ± 26.7 kPa·s) is significantly lower than that of both normal tissue (252 ± 134 kPa·s) and tumors (349 ± 222 kPa·s; p < .005). To mimic this remodeling behavior, PDAC and TAS cells were isolated from human PDAC tumors. Conditioned medium from PDAC cells was used to culture TAS-embedded collagen hydrogels. After 7 days, TAS-embedded gels in control medium reached SSM (1.45 ± 0.12 kPa) near normal pancreas, while gels maintained with conditioned medium achieved higher SSM (3.38 ± 0.146 kPa) consistent with tumors. Taken together, we have demonstrated an in vitro system that recapitulates in vivo stiffening of PDAC tumors. In addition, our quantification of viscoelastic properties suggests that elastography algorithms incorporating viscosity may be able to more accurately distinguish between pancreatic cancer and pancreatitis.

STATEMENT OF SIGNIFICANCE

Understanding tumor-stroma crosstalk in pancreatic ductal adenocarcinoma (PDAC) is challenged by a lack of stroma-mimicking model systems. To design appropriate models, pancreatic tissue must be characterized with a method capable of evaluating in vitro models as well. Our indentation-based characterization tool quantified the distinct viscoelastic signatures of inflamed resections from pancreatitis, tumors from PDAC, and otherwise normal tissue to inform development of mechanically appropriate engineered tissues and scaffolds. We also made progress toward a 3D in vitro system that recapitulates mechanical properties of tumors. Our in vitro model of stromal cells in collagen and complementary characterization system can be used to investigate mechanisms of cancer-stroma crosstalk in PDAC and to propose and test innovative therapies.

AFM-based detection of glycocalyx degradation and endothelial stiffening in the db/db mouse model of diabetes

Degradation of the glycocalyx and stiffening of endothelium are important pathophysiological components of endothelial dysfunction. However, to our knowledge, these events have not been investigated in tandem in experimental diabetes. Here, the mechanical properties of the glycocalyx and endothelium in ex vivo mouse aorta were determined simultaneously in indentation experiments with an atomic force microscope (AFM) for diabetic db/db and control db/+ mice at ages of 11–19 weeks. To analyze highly heterogeneous aorta samples, we developed a tailored classification procedure of indentation data based on a bi-layer brush model supplemented with Hertz model for quantification of nanomechanics of endothelial regions with and without the glycocalyx surface. In db/db mice, marked endothelial stiffening and reduced glycocalyx coverage were present already in 11-week-old mice and persisted in older animals. In contrast, reduction of the effective glycocalyx length was progressive and was most pronounced in 19-week-old db/db mice. The reduction of the glycocalyx length correlated with an increasing level of glycated haemoglobin and decreased endothelial NO production. In conclusion, AFM nanoindentation analysis revealed that stiffening of endothelial cells and diminished glycocalyx coverage occurred in early diabetes and were followed by the reduction of the glycocalyx length that correlated with diabetes progression.

Apical surface supracellular mechanical properties in polarized epithelium using noninvasive acoustic force spectroscopy

Maintenance of epithelial tissue integrity requires coordination between cell–cell adherens junctions, tight junctions (TJ), and the perijunctional actomyosin cytoskeleton. Here we addressed the hypothesis that alterations in TJ structure and remodeling of the actomyosin cytoskeleton modify epithelial mechanics. Current methods to measure supracellular mechanical properties disrupt intact monolayers, therefore, we developed a novel method using noncontact acoustic frequency-modulation atomic force microscopy (FM-AFM) and tested it on MDCK polarized monolayers. Our results show that double knockdown (dKD) of ZO-1/ZO-2 elevates the apical epithelial tension and effective viscosity. Interestingly, epithelial tension is more sensitive to inhibition of myosin II ATPase activity than to inhibition of ROCK activity, but viscosity is highly sensitive to both. Additionally, we showed epithelial intercellular pulling forces at tricellular junctions and adhesion forces in dKD cells are elevated with an increase in contractility. In conclusion, FM-AFM enables the physiological and quantitative investigation of mechanics in intact epithelium.

Determination of apical tension, fluidity, and intercellular adhesive forces in an epithelial monolayer are currently disruptive. Here the authors present a method using acoustic force microscopy to measure changes in these parameters upon tight junction structural alterations in a MDCK monolayer.

Atomic Force Microscopy in Characterizing Cell Mechanics for Biomedical Applications: A Review

Cell mechanics is a novel label-free biomarker for indicating cell states and pathological changes. The advent of atomic force microscopy (AFM) provides a powerful tool for quantifying the mechanical properties of single living cells in aqueous conditions. The wide use of AFM in characterizing cell mechanics in the past two decades has yielded remarkable novel insights in understanding the development and progression of certain diseases, such as cancer, showing the huge potential of cell mechanics for practical applications in the field of biomedicine. In this paper, we reviewed the utilization of AFM to characterize cell mechanics. First, the principle and method of AFM single-cell mechanical analysis was presented, along with the mechanical responses of cells to representative external stimuli measured by AFM. Next, the unique changes of cell mechanics in two types of physiological processes (stem cell differentiation, cancer metastasis) revealed by AFM were summarized. After that, the molecular mechanisms guiding cell mechanics were analyzed. Finally the challenges and future directions were discussed.