Spherical indentation of soft matter beyond the Hertzian regime: numerical and experimental validation of hyperelastic models

The lack of practicable nonlinear elastic contact models frequently compels the inappropriate use of Hertzian models in analyzing indentation data and likely contributes to inconsistencies associated with the results of biological atomic force microscopy measurements. We derived and validated with the aid of the finite element method force-indentation relations based on a number of hyperelastic strain energy functions. The models were applied to existing data from indentation, using microspheres as indenters, of synthetic rubber-like gels, native mouse cartilage tissue, and engineered cartilage. For the biological tissues, the Fung and single-term Ogden models achieved the best fits of the data while all tested hyperelastic models produced good fits for the synthetic gels. The Hertz model proved to be acceptable for the synthetic gels at small deformations (strain < 0.05 for the samples tested), but not for the biological tissues. Although this finding supports the generally accepted view that many soft materials can be assumed to be linear elastic at small deformations, the nonlinear models facilitate analysis of intrinsically nonlinear tissues and large-strain indentation behavior.     

Changes in the hyperelastic properties of endothelial cells induced by tumor necrosis factor-alpha

Mechanical properties of living cells can be determined using atomic force microscopy (AFM). In this study, a novel analysis was developed to determine the mechanical properties of adherent monolayers of pulmonary microvascular endothelial cells (ECs) using AFM and finite element modeling, which considers both the finite thickness of ECs and their nonlinear elastic properties, as well as the large strain induced by AFM. Comparison of this model with the more traditional Hertzian model, which assumes linear elastic behavior, small strains, and infinite cell thickness, suggests that the new analysis can predict the mechanical response of ECs during AFM indentation better than Hertz's model, especially when using force-displacement data obtained from large indentations (>100 nm). The shear moduli and distensibility of ECs were greater when using small indentations (<100 nm) compared to large indentations (>100 nm). Tumor necrosis factor-α induced changes in the mechanical properties of ECs, which included a decrease in the average shear moduli that occurred in all regions of the ECs and an increase in distensibility in the central regions when measured using small indentations. These changes can be modeled as changes in a chain network structure within the ECs.     

Probing elasticity and adhesion of live cells by atomic force microscopy indentation

Atomic force microscopy (AFM) indentation has become an important technique for quantifying the mechanical properties of live cells at nanoscale. However, determination of cell elasticity modulus from the force–displacement curves measured in the AFM indentations is not a trivial task. The present work shows that these force–displacement curves are affected by indenter-cell adhesion force, while the use of an appropriate indentation model may provide information on the cell elasticity and the work of adhesion of the cell membrane to the surface of the AFM probes. A recently proposed indentation model (Sirghi, Rossi in Appl Phys Lett 89:243118, 2006), which accounts for the effect of the adhesion force in nanoscale indentation, is applied to the AFM indentation experiments performed on live cells with pyramidal indenters. The model considers that the indentation force equilibrates the elastic force of the cell cytoskeleton and the adhesion force of the cell membrane. It is assumed that the indenter-cell contact area and the adhesion force decrease continuously during the unloading part of the indentation (peeling model). Force–displacement curves measured in indentation experiments performed with silicon nitride AFM probes with pyramidal tips on live cells (mouse fibroblast Balb/c3T3 clone A31-1-1) in physiological medium at 37°C agree well with the theoretical prediction and are used to determine the cell elasticity modulus and indenter-cell work of adhesion. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Nonlinear compliance of elastic layers to indentation

Thin samples adherent to a rigid substrate are considerably less compliant to indentation when compared to specimens that are not geometrically confined. Analytical corrections to this so-called substrate effect exist for various types of indenters but are not applicable when large deformations are possible, as is the case in biological materials. To overcome this limitation, we construct a nonlinear scaling model characterized by one single exponent, which we explore employing a parametric finite element analysis. The model is based on asymptotes of two length scales in relation to the sample thickness, i.e., indentation depth and radius of the contact area. For small indentation depth, we require agreement with analytical, linear models, whereas for large indentation depth and extensive contact area, we recognize similarity to uniaxial deformation, indicating a divergent force required to indent nonlinear materials. In contrast, we find linear materials not to be influenced by the substrate effect beyond first order, implying that nonlinear effects originating from either the material or geometric confinement are clearly separated only in thin samples. Furthermore, in this regime the scaling model can be derived by following a heuristic argument extending a linear model to large indentation depths. Lastly, in a large indentation setting where the contact is small in comparison with sample thickness, we observe nonlinear effects independent of material type that we attribute to a higher-order influence of geometrical confinement. In this regime, we define a scalar as the ratio of strains along principal axes as obtained by comparison with the case of a point force on a half-space. We find this scalar to be in quantitative agreement with the scaling exponent, indicating an approach to distinguish between nonlinear effects in the scaling model. While we conjecture our findings to be applicable to other flat-ended indenters, we focus on the case of a flat-ended cylinder in normal contact with a thin layer. The analytical solution for small indentation associated with this problem has been given by Hayes et al. (J Biomech 5:541–551, 1972), for which we provide a convenient numerical implementation.     

Indentation and adhesive probing of a cell membrane with AFM: theoretical model and experiments

In probing adhesion and cell mechanics by atomic force microscopy (AFM), the mechanical properties of the membrane have an important if neglected role. Here we theoretically model the contact of an AFM tip with a cell membrane, where direct motivation and data are derived from a prototypical ligand-receptor adhesion experiment. An AFM tip is functionalized with a prototypical ligand, SIRPalpha, and then used to probe its native receptor on red cells, CD47. The interactions prove specific and typical in force, and also show in detachment, a sawtooth-shaped disruption process that can extend over hundreds of nm. The theoretical model here that accounts for both membrane indentation as well as membrane extension in tip retraction incorporates membrane tension and elasticity as well as AFM tip geometry and stochastic disruption. Importantly, indentation depth proves initially proportional to membrane tension and does not follow the standard Hertz model. Computations of detachment confirm nonperiodic disruption with membrane extensions of hundreds of nm set by membrane tension. Membrane mechanical properties thus clearly influence AFM probing of cells, including single molecule adhesion experiments.     

A thin-layer model for viscoelastic, stress-relaxation testing of cells using atomic force microscopy: do cell properties reflect metastatic potential?

Atomic force microscopy has rapidly become a valuable tool for quantifying the biophysical properties of single cells. The interpretation of atomic force microscopy-based indentation tests, however, is highly dependent on the use of an appropriate theoretical model of the testing configuration. In this study, a novel, thin-layer viscoelastic model for stress relaxation was developed to quantify the mechanical properties of chondrosarcoma cells in different configurations to examine the hypothesis that viscoelastic properties reflect the metastatic potential and invasiveness of the cell using three well-characterized human chondrosarcoma cell lines (JJ012, FS090, 105KC) that show increasing chondrocytic differentiation and decreasing malignancy, respectively. Single-cell stress relaxation tests were conducted at 2 h and 2 days after plating to determine cell mechanical properties in either spherical or spread morphologies and analyzed using the new theoretical model. At both time points, JJ012 cells had the lowest moduli of the cell lines examined, whereas FS090 typically had the highest. At 2 days, all cells showed an increase in stiffness and a decrease in apparent viscosity compared to the 2-h time point. Fluorescent labeling showed that the F-actin structure in spread cells was significantly different between FS090 cells and JJ012/105KC cells. Taken together with results of previous studies, these findings indicate that cell transformation and tumorigenicity are associated with a decrease in cell modulus and apparent viscosity, suggesting that cell mechanical properties may provide insight into the metastatic potential and invasiveness of a cell.     

Analysis of indentation: implications for measuring mechanical properties with atomic force microscopy.

Indentation using the atomic force microscope (AFM) has potential to measure detailed micromechanical properties of soft biological samples. However, interpretation of the results is complicated by the tapered shape of the AFM probe tip, and its small size relative to the depth of indentation. Finite element models (FEMs) were used to examine effects of indentation depth, tip geometry, and material nonlinearity and heterogeneity on the finite indentation response. Widely applied infinitesimal strain models agreed with FEM results for linear elastic materials, but yielded substantial errors in the estimated properties for nonlinear elastic materials. By accounting for the indenter geometry to compute an apparent elastic modulus as a function of indentation depth, nonlinearity and heterogeneity of material properties may be identified. Furthermore, combined finite indentation and biaxial stretch may reveal the specific functional form of the constitutive law--a requirement for quantitative estimates of material constants to be extracted from AFM indentation data.     

Mechanical double layer model for Saccharomyces Cerevisiae cell wall

The elastic modulus of the Baker’s yeast (Saccharomyces cerevisiae) cell wall reported in studies using atomic force microscopy (AFM) is two orders of magnitude lower than that obtained using whole cell compression by micromanipulation. Using finite element modelling, it is shown that Hertz-Sneddon analysis cannot be applied to AFM indentation data for single layer core–shell structures. In addition, the Reissner solution for shallow homogeneous spheres is not appropriate for thick walls such as those of yeast cells. In order to explain yeast compression measurements at different length scales, a double layer wall model is presented considering a soft external layer composed of mannoproteins, and a stiff inner layer of β-glucan fibres and chitin. Under this model, previous AFM studies using sharp indenters provide reasonable estimates of the external layer elastic modulus, while micromanipulation provides the total stiffness of the cell wall. Data from both measurements are combined to estimate the mechanical properties of the inner stiff layer.     

Stiffness Tomography by Atomic Force Microscopy

The atomic force microscope is a convenient tool to probe living samples at the nanometric scale. Among its numerous capabilities, the instrument can be operated as a nano-indenter to gather information about the mechanical properties of the sample. In this operating mode, the deformation of the cantilever is displayed as a function of the indentation depth of the tip into the sample. Fitting this curve with different theoretical models permits us to estimate the Young's modulus of the sample at the indentation spot. We describe what to our knowledge is a new technique to process these curves to distinguish structures of different stiffness buried into the bulk of the sample. The working principle of this new imaging technique has been verified by finite element models and successfully applied to living cells.     

Effect of membrane stiffness and cytoskeletal element density on mechanical stimuli within cells: an analysis of the consequences of ageing in cells

A finite element model of a single cell was created and used to compute the biophysical stimuli generated within a cell under mechanical loading. Major cellular components were incorporated in the model: the membrane, cytoplasm, nucleus, microtubules, actin filaments, intermediate filaments, nuclear lamina and chromatin. The model used multiple sets of tensegrity structures. Viscoelastic properties were assigned to the continuum components. To corroborate the model, a simulation of atomic force microscopy indentation was performed and results showed a force/indentation simulation with the range of experimental results. A parametric analysis of both increasing membrane stiffness (thereby modelling membrane peroxidation with age) and decreasing density of cytoskeletal elements (thereby modelling reduced actin density with age) was performed. Comparing normal and aged cells under indentation predicts that aged cells have a lower membrane area subjected to high strain as compared with young cells, but the difference, surprisingly, is very small and may not be measurable experimentally. Ageing is predicted to have a more significant effect on strain deep in the nucleus. These results show that computation of biophysical stimuli within cells are achievable with single-cell computational models; correspondence between computed and measured force/displacement behaviours provides a high-level validation of the model. Regarding the effect of ageing, the models suggest only small, although possibly physiologically significant, differences in internal biophysical stimuli between normal and aged cells.     

Stiffness and heterogeneity of the pulmonary endothelial glycocalyx measured by atomic force microscopy

The mechanical properties of endothelial glycocalyx were studied using atomic force microscopy with a silica bead (diameter ～18 μm) serving as an indenter. Even at indentations of several hundred nanometers, the bead exerted very low compressive pressures on the bovine lung microvascular endothelial cell (BLMVEC) glycocalyx and allowed for an averaging of stiffness in the bead-cell contact area. The elastic modulus of BLMVEC glycocalyx was determined as a pointwise function of the indentation depth before and after enzymatic degradation of specific glycocalyx components. The modulus-indentation depth profiles showed the cells becoming progressively stiffer with increased indentation. Three different enzymes were used: heparinases III and I and hyaluronidase. The main effects of heparinase III and hyaluronidase enzymes were that the elastic modulus in the cell junction regions increased more rapidly with the indentation than in BLMVEC controls, and that the effective thickness of glycocalyx was reduced. Cytochalasin D abolished the modulus increase with the indentation. The confocal profiling of heparan sulfate and hyaluronan with atomic force microscopy indentation data demonstrated marked heterogeneity of the glycocalyx composition between cell junctions and nuclear regions.     

A nonlinear biphasic fiber-reinforced porohyperviscoelastic model of articular cartilage incorporating fiber reorientation and dispersion

A nonlinear biphasic fiber-reinforced porohyperviscoelastic (BFPHVE) model of articular cartilage incorporating fiber reorientation effects during applied load was used to predict the response of ovine articular cartilage at relatively high strains (20%). The constitutive material parameters were determined using a coupled finite element-optimization algorithm that utilized stress relaxation indentation tests at relatively high strains. The proposed model incorporates the strain-hardening, tension-compression, permeability, and finite deformation nonlinearities that inherently exist in cartilage, and accounts for effects associated with fiber dispersion and reorientation and intrinsic viscoelasticity at relatively high strains. A new optimization cost function was used to overcome problems associated with large peak-to-peak differences between the predicted finite element and experimental loads that were due to the large strain levels utilized in the experiments. The optimized material parameters were found to be insensitive to the initial guesses. Using experimental data from the literature, the model was also able to predict both the lateral displacement and reaction force in unconfined compression, and the reaction force in an indentation test with a single set of material parameters. Finally, it was demonstrated that neglecting the effects of fiber reorientation and dispersion resulted in poorer agreement with experiments than when they were considered. There was an indication that the proposed BFPHVE model, which includes the intrinsic viscoelasticity of the nonfibrillar matrix (proteoglycan), might be used to model the behavior of cartilage up to relatively high strains (20%). The maximum percentage error between the indentation force predicted by the FE model using the optimized material parameters and that measured experimentally was 3%.     

Cellular Pressure and Volume Regulation and Implications for Cell Mechanics

In eukaryotic cells, small changes in cell volume can serve as important signals for cell proliferation, death, and migration. Volume and shape regulation also directly impacts the mechanics of cells and tissues. Here, we develop a mathematical model of cellular volume and pressure regulation, incorporating essential elements such as water permeation, mechanosensitive channels, active ion pumps, and active stresses in the cortex. The model can fully explain recent experimental data, and it predicts cellular volume and pressure for several models of cell cortical mechanics. Moreover, we show that when cells are subjected to an externally applied load, such as in an atomic force microscopy indentation experiment, active regulation of volume and pressure leads to a complex cellular response. Instead of the passive mechanics of the cortex, the observed cell stiffness depends on several factors working together. This provides a mathematical explanation of rate-dependent response of cells under force.     

Cellular Pressure and Volume Regulation and Implications for Cell Mechanics

In eukaryotic cells, small changes in cell volume can serve as important signals for cell proliferation, death, and migration. Volume and shape regulation also directly impacts the mechanics of cells and tissues. Here, we develop a mathematical model of cellular volume and pressure regulation, incorporating essential elements such as water permeation, mechanosensitive channels, active ion pumps, and active stresses in the cortex. The model can fully explain recent experimental data, and it predicts cellular volume and pressure for several models of cell cortical mechanics. Moreover, we show that when cells are subjected to an externally applied load, such as in an atomic force microscopy indentation experiment, active regulation of volume and pressure leads to a complex cellular response. Instead of the passive mechanics of the cortex, the observed cell stiffness depends on several factors working together. This provides a mathematical explanation of rate-dependent response of cells under force.     

Surface structure and nanomechanical properties of Shewanella putrefaciens bacteria at two pH values (4 and 10) determined by atomic force microscopy

The nanomechanical properties of gram-negative bacteria (Shewanella putrefaciens) were investigated in situ in aqueous solutions at two pH values, specifically, 4 and 10, by atomic force microscopy (AFM). For both pH values, the approach force curves exhibited subsequent nonlinear and linear regimens that were related to the progressive indentation of the AFM tip in the bacterial cell wall, including a priori polymeric fringe (nonlinear part), while the linear part was ascribed to compression of the plasma membrane. These results indicate the dynamic of surface ultrastructure in response to changes in pH, leading to variations in nanomechanical properties, such as the Young's modulus and the bacterial spring constant.     

Slow stress propagation in adherent cells

Mechanical cues influence a wide range of cellular behaviors including motility, differentiation, and tumorigenesis. Although previous studies elucidated the role of specific players such as ion channels and focal adhesions as local mechanosensors, the investigation of how mechanical perturbations propagate across the cell is necessary to understand the spatial coordination of cellular processes. Here we quantify the magnitude and timing of intracellular stress propagation, using atomic force microscopy and particle tracking by defocused fluorescence microscopy. The apical cell surface is locally perturbed by atomic force microscopy cantilever indentation, and distal displacements are measured in three dimensions by tracking integrin-bound fluorescent particles. We observe an immediate response and slower equilibration, occurring over times that increase with distance from perturbation. This distance-dependent equilibration occurs over several seconds and can be eliminated by disruption of the actin cytoskeleton. Our experimental results are not explained by traditional viscoelastic models of cell mechanics, but they are consistent with predictions from poroelastic models that include both cytoskeletal deformation and flow of the cytoplasm. Our combined atomic force microscopy-particle tracking measurements provide direct evidence of slow, distance-dependent dissipative stress propagation in response to external mechanical cues and offer new insights into mechanical models and physiological behaviors of adherent cells.     

If Cell Mechanics Can Be Described by Elastic Modulus: Study of Different Models and Probes Used in Indentation Experiments

Here we investigated the question whether cells, being highly heterogeneous objects, could be described with the elastic modulus (effective Young’s modulus) in a self-consistent way. We performed a comparative analysis of the elastic modulus derived from the indentation data obtained with atomic force microscopy (AFM) on human cervical epithelial cells (both normal and cancerous). Both sharp (cone) and dull (2500-nm radius sphere) AFM probes were used. The indentation data were processed through different elastic models. The cell was approximated as a homogeneous elastic medium that had either 1), smooth hemispherical boundary (Hertz/Sneddon models) or 2), the boundary covered with a layer of glycocalyx and membrane protrusions (“brush” models). Consistency of these approximations was investigated. Specifically, we tested the independence of the elastic modulus of the indentation depth, which is assumed in these models. We demonstrated that only one model showed consistency in treating cells as a homogeneous elastic medium, namely, the brush model, when processing the indentation data collected with the dull AFM probe. The elastic modulus demonstrated strong depth dependence in all models: Hertz/Sneddon models (no brush taken into account), and when the brush model was applied to the data collected with sharp conical probes. We conclude that it is possible to describe the elastic properties of the cell body by means of an effective elastic modulus, used in a self-consistent way, when using the brush model to analyze data collected with a dull AFM probe. The nature of these results is discussed.     

If Cell Mechanics Can Be Described by Elastic Modulus: Study of Different Models and Probes Used in Indentation Experiments

Here we investigated the question whether cells, being highly heterogeneous objects, could be described with the elastic modulus (effective Young’s modulus) in a self-consistent way. We performed a comparative analysis of the elastic modulus derived from the indentation data obtained with atomic force microscopy (AFM) on human cervical epithelial cells (both normal and cancerous). Both sharp (cone) and dull (2500-nm radius sphere) AFM probes were used. The indentation data were processed through different elastic models. The cell was approximated as a homogeneous elastic medium that had either 1), smooth hemispherical boundary (Hertz/Sneddon models) or 2), the boundary covered with a layer of glycocalyx and membrane protrusions (“brush” models). Consistency of these approximations was investigated. Specifically, we tested the independence of the elastic modulus of the indentation depth, which is assumed in these models. We demonstrated that only one model showed consistency in treating cells as a homogeneous elastic medium, namely, the brush model, when processing the indentation data collected with the dull AFM probe. The elastic modulus demonstrated strong depth dependence in all models: Hertz/Sneddon models (no brush taken into account), and when the brush model was applied to the data collected with sharp conical probes. We conclude that it is possible to describe the elastic properties of the cell body by means of an effective elastic modulus, used in a self-consistent way, when using the brush model to analyze data collected with a dull AFM probe. The nature of these results is discussed.     

Influence of nonuniform geometry on nanoindentation of viral capsids

A series of recent nanoindentation experiments on the protein shells (capsids) of viruses has established atomic force microscopy (AFM) as a useful framework for probing the mechanics of large protein assemblies. Specifically these experiments provide an opportunity to study the coupling of the global assembly response to local conformational changes. AFM experiments on cowpea chlorotic mottle virus, known to undergo a pH-controlled swelling conformational change, have revealed a pH-dependent mechanical response. Previous theoretical studies have shown that homogeneous changes in shell geometry can play a significant role in the mechanical response. This article develops a method for accurately capturing the heterogeneous geometry of a viral capsid and explores its effect on mechanical response with a nonlinear continuum elasticity model. Models of both native and swollen cowpea chlorotic mottle virus capsids are generated from x-ray crystal structures, and are used in finite element simulations of AFM indentation along two-, three-, and fivefold icosahedral symmetry orientations. The force response of the swollen capsid model is observed to be softer by roughly a factor of two, significantly more nonlinear, and more orientation-dependent than that of a native capsid with equivalent elastic moduli, demonstrating that capsid geometric heterogeneity can have significant effects on the global structural response.     

Modeling and simulation of the mechanical response from nanoindentation test of DNA-filled viral capsids

Viruses can be described as biological objects composed mainly of two parts: a stiff protein shell called a capsid, and a core inside the capsid containing the nucleic acid and liquid. In many double-stranded DNA bacterial viruses (aka phage), the volume ratio between the liquid and the encapsidated DNA is approximately 1:1. Due to the dominant DNA hydration force, water strongly mediates the interaction between the packaged DNA strands. Therefore, water that hydrates the DNA plays an important role in nanoindentation experiments of DNA-filled viral capsids. Nanoindentation measurements allow us to gain further insight into the nature of the hydration and electrostatic interactions between the DNA strands. With this motivation, a continuum-based numerical model for simulating the nanoindentation response of DNA-filled viral capsids is proposed here. The viral capsid is modeled as large- strain isotropic hyper-elastic material, whereas porous elasticity is adopted to capture the mechanical response of the filled viral capsid. The voids inside the viral capsid are assumed to be filled with liquid, which is modeled as a homogenous incompressible fluid. The motion of a fluid flowing through the porous medium upon capsid indentation is modeled using Darcy’s law, describing the flow of fluid through a porous medium. The nanoindentation response is simulated using three-dimensional finite element analysis and the simulations are performed using the finite element code Abaqus. Force-indentation curves for empty, partially and completely DNA-filled capsids are directly compared to the experimental data for bacteriophage λ. Material parameters such as Young’s modulus, shear modulus, and bulk modulus are determined by comparing computed force-indentation curves to the data from the atomic force microscopy (AFM) experiments. Predictions are made for pressure distribution inside the capsid, as well as the fluid volume ratio variation during the indentation test.