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

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.

Reliable, standardized measurements for cell mechanical properties

Atomic force microscopy (AFM) has become indispensable for studying biological and medical samples. More than two decades of experiments have revealed that cancer cells are softer than healthy cells (for measured cells cultured on stiff substrates). The softness or, more precisely, the larger deformability of cancer cells, primarily independent of cancer types, could be used as a sensitive marker of pathological changes. The wide application of biomechanics in clinics would require designing instruments with specific calibration, data collection, and analysis procedures. For these reasons, such development is, at present, still very limited, hampering the clinical exploitation of mechanical measurements. Here, we propose a standardized operational protocol (SOP), developed within the EU ITN network Phys2BioMed, which allows the detection of the biomechanical properties of living cancer cells regardless of the nanoindentation instruments used (AFMs and other indenters) and the laboratory involved in the research. We standardized the cell cultures, AFM calibration, measurements, and data analysis. This effort resulted in a step-by-step SOP for cell cultures, instrument calibration, measurements, and data analysis, leading to the concordance of the results (Young's modulus) measured among the six EU laboratories involved. Our results highlight the importance of the SOP in obtaining a reproducible mechanical characterization of cancer cells and paving the way toward exploiting biomechanics for diagnostic purposes in clinics.

Quantifying the Viscosity of Individual Submicrometer Semisolid Particles Using Atomic Force Microscopy

Atmospheric aerosols' viscosities can vary significantly depending on their composition, mixing states, relative humidity (RH) and temperature. The diffusion time scale of atmospheric gases into an aerosol is largely governed by its viscosity, which in turn influences heterogeneous chemistry and climate-relevant aerosol effects. Quantifying the viscosity of aerosols in the semisolid phase state is particularly important as they are prevalent in the atmosphere and have a wide range of viscosities. Currently, direct viscosity measurements of submicrometer individual atmospheric aerosols are limited, largely due to the inherent size limitations of existing experimental techniques. Herein, we present a method that utilizes atomic force microscopy (AFM) to directly quantify the viscosity of substrate-deposited individual submicrometer semisolid aerosol particles as a function of RH. The method is based on AFM force spectroscopy measurements coupled with the Kelvin-Voigt viscoelastic model. Using glucose, sucrose, and raffinose as model systems, we demonstrate the accuracy of the AFM method within the viscosity range of ∼104-107 Pa s. The method is applicable to individual particles with sizes ranging from tens of nanometers to several micrometers. Furthermore, the method does not require prior knowledge on the composition of studied particles. We anticipate future measurements utilizing the AFM method on atmospheric aerosols at various RH to aid in our understanding of the range of aerosols' viscosities, the extent of particle-to-particle viscosity variability, and how these contribute to the particle diversity observable in the atmosphere.

Insights and guidelines to interpret forces and deformations at the nanoscale by using a tapping mode AFM simulator: dForce 2.0

Amplitude modulation (tapping mode) AFM is the most versatile AFM mode for imaging surfaces at the nanoscale in air and liquid environments. However, it remains challenging to estimate the forces and deformations exerted by the tip. We introduce a new simulator environment to predict the values of the observables in tapping mode AFM experiments. The relevant feature of dForce 2.0 is the incorporation of contact mechanics models aimed to describe the properties of ultrathin samples. These models were essential to determine the forces applied on samples such as proteins, self-assembled monolayers, lipid bilayers, and few-layered materials. The simulator incorporates two types of long-range magnetic forces. The simulator is written in an open-source code (Python) and it can be run from a personal computer.

The cell softening as a universal indicator of cell damage during cytotoxic effects

Cytotoxicity assays are essential tests in studies on the safety and biocompatibility of various substances and on the efficiency of anticancer drugs. The most frequently used assays commonly require application of externally added labels and read only collective response of cells. Recent studies show that the internal biophysical parameters of cells can be associated with the cellular damage. Therefore, using atomic force microscopy, we assessed the changes in the viscoelastic parameters of cells treated with eight different common cytotoxic agents to gain a more systematic view of the occurring mechanical changes. With the robust statistical analysis to account for both the cell-level variability and the experimental reproducibility, we have found that cell softening is a common response after each treatment. More precisely, the combined changes in the viscoelastic parameters of power-law rheology model led to a significant decrease of the apparent elastic modulus. The comparison with the morphological parameters (cytoskeleton and cell shape) demonstrated a higher sensitivity of the mechanical parameters versus the morphological ones. The obtained results support the idea of cell mechanics-based cytotoxicity tests and suggest a common way of a cell responding to damaging actions by softening.

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.

Determining Spatial Variability of Elastic Properties for Biological Samples Using AFM

Measuring the mechanical properties (i.e., elasticity in terms of Young's modulus) of biological samples using Atomic Force Microscopy (AFM) indentation at the nanoscale has opened new horizons in studying and detecting various pathological conditions at early stages, including cancer and osteoarthritis. It is expected that AFM techniques will play a key role in the future in disease diagnosis and modeling using rigorous mathematical criteria (i.e., automated user-independent diagnosis). In this review, AFM techniques and mathematical models for determining the spatial variability of elastic properties of biological materials at the nanoscale are presented and discussed. Significant issues concerning the rationality of the elastic half-space assumption, the possibility of monitoring the depth-dependent mechanical properties, and the construction of 3D Young's modulus maps are also presented.

Sensitivity of viscoelastic characterization in multi-harmonic atomic force microscopy

Quantifying the nanomechanical properties of soft-matter using multi-frequency atomic force microscopy (AFM) is crucial for studying the performance of polymers, ultra-thin coatings, and biological systems. Such characterization processes often make use of cantilever's spectral components to discern nanomechanical properties within a multi-parameter optimization problem. This could inadvertently lead to an over-determined parameter estimation with no clear relation between the identified parameters and their influence on the experimental data. In this work, we explore the sensitivity of viscoelastic characterization in polymeric samples to the experimental observables of multi-frequency intermodulation AFM. By performing simulations and experiments we show that surface viscoelasticity has negligible effect on the experimental data and can lead to inconsistent and often non-physical identified parameters. Our analysis reveals that this lack of influence of the surface parameters relates to a vanishing gradient and non-convexity while minimizing the objective function. By removing the surface dependency from the model, we show that the characterization of bulk properties can be achieved with ease and without any ambiguity. Our work sheds light on the sensitivity issues that can be faced when optimizing for a large number of parameters and observables in AFM operation, and calls for the development of new viscoelastic models at the nanoscale and improved computational methodologies for nanoscale mapping of viscoelasticity using AFM.

Building a tissue: mesenchymal and epithelial cell spheroids' mechanical properties at micro- and nanoscale

Cell transitions between the epithelial and mesenchymal phenotypes provide the regulated morphogenesis and regeneration throughout the ontogenesis. The tissue mechanics and mechanotransduction play an essential role in these processes. Cell spheroids reproduce the cell density of native tissues and represent simple building blocks for the tissue engineering purposes. The mechanical properties of mesenchymal and epithelial cells have been extensively studied in 2D monolayer cultures, but have not been sufficiently compared in spheroids. Here, we have simultaneously applied several techniques to assess the mechanical parameters of such spheroids. The local surface mechanical properties were measured by AFM, and the bulk properties were analyzed with parallel-plate compression, as well as by observing cut opening after microdissection. The comparison of the collected data allowed us to apply the model of a solid body with surface tension, and estimate the parameters of this model. We found an expectedly higher surface tension in mesenchymal spheroids, as well as a higher bulk modulus and relaxation time. The two latter parameters agree with the bulk poroelastic behavior of spheroids, and with the higher cell density and extracellular matrix content in mesenchymal spheroids. The higher tension of the surface layer cells in mesenchymal cell spheroids was also confirmed by the viscoelastic AFM characterization. The cell phenotype affected the self-organization during the spheroid formation, as well as the structure, biomechanical properties, and spreading of spheroids. The obtained results will contribute to a more detailed description of spheroid and tissue biomechanics, and will help in controlling the tissue regeneration and morphogenesis. STATEMENT OF SIGNIFICANCE: Spheroids are widely used as building blocks for scaffold-based and scaffold-free strategies in tissue engineering. In the majority of the past studies, either the concept of a solid body or a liquid with surface tension was used to describe the biomechanical behavior of spheroids. Here, we have used a model which combines both aspects, a solid body with surface tension. The "solid" aspect was described as a visco-poroelastic material, affected by the liquid redistribution through the cells and ECM at the scale of the whole spheroid. A higher surface tension was found for mesenchymal spheroids than that for epithelial spheroids, observed as a higher stiffness of the spheroid surface, as well as a larger spontaneous opening of the cut edges after microdissection.

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.

A Continuum-Tensegrity Computational Model for Chondrocyte Biomechanics in AFM Indentation and Micropipette Aspiration

Mechanical stimuli are fundamental in the development of organs and tissues, their growth, regeneration or disease. They influence the biochemical signals produced by the cells, and, consequently, the development and spreading of a disease. Moreover, tumour cells are usually characterized by a decrease in the cell mechanical properties that may be directly linked to their metastatic potential. Thus, recently, the experimental and computational study of cell biomechanics is facing a growing interest. Various experimental approaches have been implemented to describe the passive response of cells; however, cell variability and complex experimental procedures may affect the obtained mechanical properties. For this reason, in-silico computational models have been developed through the years, to overcome such limitations, while proposing valuable tools to understand cell mechanical behaviour. This being the case, we propose a combined continuous-tensegrity finite element (FE) model to analyse the mechanical response of a cell and its subcomponents, observing how every part contributes to the overall mechanical behaviour. We modelled both Atomic Force Microscopy (AFM) indentation and micropipette aspiration techniques, as common mechanical tests for cells and elucidated also the role of cell cytoplasm and cytoskeleton in the global cell mechanical response.

Accurate Wide-Modulus-Range Nanomechanical Mapping of Ultrathin Interfaces with Bimodal Atomic Force Microscopy

The nanoscale determination of the mechanical properties of interfaces is of paramount relevance in materials science and cell biology. Bimodal atomic force microscopy (AFM) is arguably the most advanced nanoscale method for mapping the elastic modulus of interfaces. Simulations, theory, and experiments have validated bimodal AFM measurements on thick samples (from micrometer to millimeter). However, the bottom-effect artifact, this is, the influence of the rigid support on the determination of the Young's modulus, questions its accuracy for ultrathin materials and interfaces (1-15 nm). Here we develop a bottom-effect correction method that yields the intrinsic Young's modulus value of a material independent of its thickness. Experiments and numerical simulations validate the accuracy of the method for a wide range of materials (1 MPa to 100 GPa). Otherwise, the Young's modulus of an ultrathin material might be overestimated by a 10-fold factor.

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.

A Cell's Viscoelasticity Measurement Method Based on the Spheroidization Process of Non-Spherical Shaped Cell

The mechanical properties of biological cells, especially the elastic modulus and viscosity of cells, have been identified to reflect cell viability and cell states. The existing measuring techniques need additional equipment or operation condition. This paper presents a cell's viscoelasticity measurement method based on the spheroidization process of non-spherical shaped cell. The viscoelasticity of porcine fetal fibroblast was measured. Firstly, we introduced the process of recording the spheroidization process of porcine fetal fibroblast. Secondly, we built the viscoelastic model for simulating a cell's spheroidization process. Then, we simulated the spheroidization process of porcine fetal fibroblast and got the simulated spheroidization process. By identifying the parameters in the viscoelastic model, we got the elasticity (500 Pa) and viscosity (10 Pa·s) of porcine fetal fibroblast. The results showed that the magnitude of the elasticity and viscosity were in agreement with those measured by traditional method. To verify the accuracy of the proposed method, we imitated the spheroidization process with silicone oil, a kind of viscous and uniform liquid with determined viscosity. We did the silicone oil's spheroidization experiment and simulated this process. The simulation results also fitted the experimental results well.

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

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

Mapping mechanical properties of living cells at nanoscale using intrinsic nanopipette-sample force interactions

Mechanical properties of living cells determined by cytoskeletal elements play a crucial role in a wide range of biological functions. However, low-stress mapping of mechanical properties with nanoscale resolution but with a minimal effect on the fragile structure of cells remains difficult. Scanning Ion-Conductance Microscopy (SICM) for quantitative nanomechanical mapping (QNM) is based on intrinsic force interactions between nanopipettes and samples and has been previously suggested as a promising alternative to conventional techniques. In this work, we have provided an alternative estimation of intrinsic force and stress and demonstrated the possibility to perform qualitative and quantitative analysis of cell nanomechanical properties of a variety of living cells. Force estimation on decane droplets with well-known elastic properties, similar to living cells, revealed that the forces applied using a nanopipette are much smaller than in the case using atomic force microscopy. We have shown that we can perform nanoscale topography and QNM using a scanning procedure with no detectable effect on live cells, allowing long-term QNM as well as detection of nanomechanical properties under drug-induced alterations of actin filaments and microtubulin.

A New Approach for the AFM-Based Mechanical Characterization of Biological Samples

The AFM nanoindentation technique is a powerful tool for the mechanical characterization of biological samples at the nanoscale. The data analysis of the experimentally obtained results is usually performed using the Hertzian contact mechanics. However, the aforementioned theory can be applied only in cases that the sample is homogeneous and isotropic and presents a linear elastic response. However, biological samples often present depth-dependent mechanical properties, and the Hertzian analysis cannot be used. Thus, in this paper, a different approach is presented, based on a new physical quantity used for the determination of the mechanical properties at the nanoscale. The aforementioned physical quantity is the work done by the indenter per unit volume. The advantages of the presented analysis are significant since the abovementioned magnitude can be used to examine if a sample can be approximated to an elastic half-space. If this approximation is valid, then the new proposed method enables the accurate calculation of Young's modulus. Additionally, it can be used to explore the mechanical properties of samples that are characterized by a depth-dependent mechanical behavior. In conclusion, the proposed analysis presents an accurate yet simple technique for the determination of the mechanical properties of biological samples at the nanoscale that can be also used beyond the Hertzian limit.

Viscoelasticity and Volume of Cortical Neurons under Glutamate Excitotoxicity and Osmotic Challenges

Neural activity depends on the maintenance of ionic and osmotic homeostasis. Under these conditions, the cell volume must be regulated to maintain optimal neural function. A disturbance in the neuronal volume regulation often occurs in pathological conditions such as glutamate excitotoxicity. The cell volume, mechanical properties, and actin cytoskeleton structure are tightly connected to achieve the cell homeostasis. Here, we studied the effects of glutamate-induced excitotoxicity, external osmotic pressure, and inhibition of actin polymerization on the viscoelastic properties and volume of neurons. Atomic force microscopy was used to map the viscoelastic properties of neurons in time-series experiments to observe the dynamical changes and a possible recovery. The data obtained on cultured rat cortical neurons were compared with the data obtained on rat fibroblasts. The neurons were found to be more responsive to the osmotic challenges but less sensitive to the inhibition of actin polymerization than fibroblasts. The alterations of the viscoelastic properties caused by glutamate excitotoxicity were similar to those induced by the hypoosmotic stress, but, in contrast to the latter, they did not recover after the glutamate removal. These data were consistent with the dynamic volume changes estimated using ratiometric fluorescent dyes. The recovery after the glutamate-induced excitotoxicity was slow or absent because of a steady increase in intracellular calcium and sodium concentrations. The viscoelastic parameters and their changes were related to such parameters as the actin cortex stiffness, tension, and cytoplasmic viscosity.

Nanomechanical properties of enucleated cells: contribution of the nucleus to the passive cell mechanics

Background

The nucleus, besides its functions in the gene maintenance and regulation, plays a significant role in the cell mechanosensitivity and mechanotransduction. It is the largest cellular organelle that is often considered as the stiffest cell part as well. Interestingly, the previous studies have revealed that the nucleus might be dispensable for some of the cell properties, like polarization and 1D and 2D migration. Here, we studied how the nanomechanical properties of cells, as measured using nanomechanical mapping by atomic force microscopy (AFM), were affected by the removal of the nucleus.

Methods

The mass enucleation procedure was employed to obtain cytoplasts (enucleated cells) and nucleoplasts (nuclei surrounded by plasma membrane) of two cell lines, REF52 fibroblasts and HT1080 fibrosarcoma cells. High-resolution viscoelastic mapping by AFM was performed to compare the mechanical properties of normal cells, cytoplasts, and nucleoplast. The absence or presence of the nucleus was confirmed with fluorescence microscopy, and the actin cytoskeleton structure was assessed with confocal microscopy.

Results

Surprisingly, we did not find the softening of cytoplasts relative to normal cells, and even some degree of stiffening was discovered. Nucleoplasts, as well as the nuclei isolated from cells using a detergent, were substantially softer than both the cytoplasts and normal cells.

Conclusions

The cell can maintain its mechanical properties without the nucleus. Together, the obtained data indicate the dominating role of the actomyosin cytoskeleton over the nucleus in the cell mechanics at small deformations inflicted by AFM.

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.

Bottom Effect in Atomic Force Microscopy Nanomechanics

In this work, the influence of the rigid substrate on the determination of the sample Young's modulus, the so-called bottom-effect artifact, is demonstrated by an atomic force microscopy force-spectroscopy experiment. The nanomechanical properties of a one-component supported lipid membrane (SLM) exhibiting areas of two different thicknesses are studied: While a standard contact mechanics model (Sneddon) provides two different elastic moduli for these two morphologies, it is shown that Garcia's bottom-effect artifact correction yields a unique value, as expected for an intrinsic material property. Remarkably, it is demonstrated that the ratio between the contact radius (and not only the indentation) and the sample thickness is the key parameter addressing the relevance of the bottom-effect artifact. The experimental results are validated by finite element method simulations providing a solid support to Garcia's theory. The amphiphilic nature of the investigated material is representative of several kinds of lipids, suggesting that the results have far reaching implications for determining the correct Young's modulus of SLMs. The generality of Garcia's bottom-effect artifact correction allows its application to every kind of supported soft film.

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.

Extracting viscoelastic material parameters using an atomic force microscope and static force spectroscopy

Atomic force microscopy (AFM) techniques have provided and continue to provide increasingly important insights into surface morphology, mechanics, and other critical material characteristics at the nanoscale. One attractive implementation involves extracting meaningful material properties, which demands physically accurate models specifically designed for AFM experimentation and simulation. The AFM community has pursued the precise quantification and extraction of rate-dependent material properties, in particular, for a significant period of time, attempting to describe the standard viscoelastic response of materials. AFM static force spectroscopy (SFS) is one approach commonly used in pursuit of this goal. It is capable of acquiring rich temporal insight into the behavior of a sample. During AFM-SFS experiments the cantilever base approaches samples with a nearly constant velocity, which is manipulated to investigate different timescales of the mechanical response. This manuscript seeks to build upon our previous work and presents an approach to extracting useful linear viscoelastic information from AFM-SFS experiments. In addition, the basis for selecting and restricting the model parameters for fitting is discussed from the perspective of applying this technique on a practical level. This work begins with a guided discussion that develops a fit function from fundamental laws, continues with conditioning a raw SFS experimental dataset, and concludes with the fit and prediction of viscoelastic response parameters such as storage modulus, loss modulus, loss angle, and compliance. These steps constitute a complete guide to leveraging AFM-SFS data to estimate key material parameters, with a series of detailed insights into both the methodology and supporting analytical choices.

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

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

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.

Subsurface Imaging of Cell Organelles by Force Microscopy

The development of high-resolution, label-free, noninvasive, and subsurface microscopy methods of living cells remains a formidable problem. Force-microscopy-based stiffness measurements contribute to our understanding of single-cell nanomechanics. The elastic properties of the cell's outer structures, such as the plasma membrane and actin cytoskeleton, dominate stiffness measurements, which in turns prevents the imaging of intracellular structures. We propose that the above limitation could be overcome by combining 2D sections of the cell's viscoelastic properties. We show the simultaneous imaging of the outer cell's cytoskeleton and the organelles inside the nucleus. The elastic component of interaction force carries information on the cell's outer elements as the cortex and the actin cytoskeleton. The inelastic component is sensitive to the hydrodynamic drag of the inner structures such the nucleoli.

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.

Acquisition of time-frequency localized mechanical properties of biofilms and single cells with high spatial resolution

Biofilms are a cluster of bacteria embedded in extracellular polymeric substances (EPS) that contain a complex composition of polysaccharides, proteins, and extracellular DNA (eDNA). Desirable mechanical properties of the biofilms are critical for their survival, propagation, and dispersal, and the response of mechanical properties to different treatment conditions also sheds light on biofilm control and eradication in vivo and on engineering surfaces. However, it is challenging yet important to investigate mechanical behaviors of biofilms with a high spatial resolution because biofilms are very heterogeneous. Moreover, biofilms are viscoelastic, and their time-dependent mechanical behavior is difficult to capture. Herein, we develop a powerful technique that combines the high spatial resolution of an atomic force microscope (AFM) with a rigorous history-dependent viscoelastic analysis to deliver highly spatial-localized biofilm properties within a wide time-frequency window. By exploiting the use of static force spectroscopy in combination with an appropriate viscoelastic framework, we highlight the intensive amount of time-dependent information experimentally available that has been largely overlooked. It is shown that this technique provides a detailed nanorheological signature of the biofilms even at the single-cell level. We share the computational routines that would allow any user to perform the analysis from experimental raw data. The detailed localization of mechanical properties in space and in time-frequency domain provides insights into the understanding of biofilm stability, cohesiveness, dispersal, and control.