Atomic Force Microscopy Indentation Experiments Research Articles

Virtual indentation of the empty capsid of the minute virus of mice using a minimal coarse-grained model.

A minimal coarse-grained model for T=1 viral capsids assembled from 20 protein rigid trimers has been designed by extending a previously proposed form of the interaction energy written as a sum of anisotropic pairwise interactions between the trimeric capsomers. The extension of the model has been performed to properly account for the coupling between two internal coordinates: the one that measures the intercapsomer distance and the other that gives the intercapsomer dihedral angle. The model has been able to fit with less than a 10% error the atomic force microscopy (AFM) indentation experimental data for the empty capsid of the minute virus of mice (MVM), providing in this way an admissible picture of the main mechanisms behind the capsid deformations. In this scenario, the bending of the intercapsomer dihedral angle is the angular internal coordinate that can support larger deformations away from its equilibrium values, determining important features of the AFM indentation experiments as the elastic constants along the three symmetry axes of the capsid and the critical indentations. From the value of one of the parameters of our model, we conclude that trimers in the MVM must be quite oblate tops, in excellent agreement with their known structure. The transition from the linear to the nonlinear regimes sampled in the indentation process appears to be an interesting topic for future research in physical virology.

Stiffening of polydimethylsiloxane surface as result of exposure to low‐pressure argon discharge plasma

AbstractPolydimethylsiloxane (PDMS) surface has been exposed to negative glow discharge plasma of a low‐pressure discharge in pure argon for a duration varying from 10 s to 2 min. Shallow atomic force microscopy indentation experiments were performed to investigate the effect of plasma treatment on PDMS surface stiffness and adhesion energy to the silicon nitride indenter surface. The results showed an important increase in Young's modulus and a slight increase of work of adhesion. X‐ray photoelectron spectroscopy investigation of plasma‐treated PDMS surface showed a decrease of atomic content of carbon accompanied by an increase of oxygen content and suggests that the stiffening of PDMS surface can be attributed to generation of new cross‐linking Si–O–Si and Si–CH2–CH2–Si bonds between PDMS molecules.

Atomic Force Microscopy Nanomechanics of Hard Nanometer-Thick Films on Soft Substrates: Insights into Stretchable Conductors

The nanomechanical properties of ultrathin and nanostructured films of rigid electronic materials on soft substrates are of crucial relevance to realize materials and devices for stretchable electronics. Of particular interest are bending deformations in buckled nanometer-thick films or patterned networks of rigid materials as they can be exploited to compensate for the missing tensile elasticity. Here, we perform atomic force microscopy indentation experiments and electrical measurements to characterize the nanomechanics of ultrathin gold films on a polydimethylsiloxane (PDMS) elastomer. The measured force-indentation data can be analyzed in terms of a simple analytical model describing a bending plate on a semi-infinite soft substrate. The resulting method enables us to quantify the local Young’s modulus of elasticity of the nanometer-thick film. Systematic variation of the gold layer thickness reveals the presence of a diffuse interface between the metal film and the elastomer substrate that does not contribute to the bending stiffness. The effect is associated with gold clusters that penetrate the silicone and are not directly connected to the ultrathin film. Only above a critical layer thickness, percolation of the metallic thin film happens, causing a linear increase in bending stiffness and electrical conductivity.

AFM Force Relaxation Curve Reveals That the Decrease of Membrane Tension Is the Essential Reason for the Softening of Cancer Cells.

Differences in stiffness constitute an extremely important aspect of the mechanical differences between cancer cells and normal cells, and atomic force microscopy (AFM) is the most commonly used tool to characterize the difference in stiffness. However, the process of mechanical characterization using AFM has been controversial and the influence of the membrane tension on AFM measurement results was often ignored. Here, a physical model involving a simultaneous consideration of the effects of the cell membrane, cytoskeleton network and cytosol was proposed. We carried out a theoretical analysis of AFM force relaxation curves, and as a result solved many of the remaining controversial issues regarding AFM-based mechanical characterization of cells, and provided a quantitative solution for the membrane tension measured using AFM indentation experiments for the first time. From the results of experiments on cells with different adherent shapes and different pairs of normal cells and cancer cells, we found additional force provided by membrane tension to be the main component of the force applied to the AFM probe, with decreased cell membrane tension being the essential reason for the greater softness of cancer cells than of normal cells. Hence, regulating membrane tension may become an important method for regulating the behavior of cancer cells.

Nanomechanical Properties of Articular Cartilage Due to the PRP Injection in Experimental Osteoarthritis in Rabbits.

The purpose of this study was twofold. Firstly, we proposed a measurement protocol for the atomic force microscopy (AFM) method to determine the nanomechanical properties of articular cartilage in experimental osteoarthritis in rabbits. Then, we verified if mechanical properties can be evaluated with AFM shortly after platelet-rich plasma (PRP) injection. We hypothesized that the modulus determined by AFM indentation experiments could be utilized as a progressive disease marker during the treatment of osteoarthritis. The rabbits were equally divided into three groups of six: control (group 1); injections of saline (0.5 mL) and 10% surgical talc (Talcum Pharmaceutical®, Minsk, Belarus) were delivered into the right knee under the patella (group 2 and 3); and PRP was injected into the right knee (group 3). In group 2, the arithmetic average of absolute values (Ra) change was a 25% increase; the maximum peak height (Rp) increased by over 102%, while the mean spacing between local peaks (S) increased by 28% (p < 0.05). In group 3, Ra increased by 14% and Rp increased by 32%, while S decreased by 75% (p < 0.05). The Young’s modulus of the surface layers decreased by 18% as a result of induced model of osteoarthritis (IMO) (p < 0.05), and it increased by 9% (p < 0.05) as a result of PRP therapy, which means that the mechanical properties of cartilage were partially recovered. This research demonstrates that Young’s modulus utilized on a nanometer scale has potential to be a progressive disease marker during the treatment of osteoarthritis.

Compression, Rupture, and Puncture of Model Membranes at the Molecular Scale.

Elastic properties of biological membranes are involved in a large number of membrane functionalities and activities. Conventionally characterized in terms of Young's modulus, bending stiffness and stretching modulus, membrane mechanics can be assessed at high lateral resolution by means of atomic force microscopy (AFM). Here we show that the mechanical response of biomimetic model systems such as supported lipid bilayers (SLBs) is highly affected by the size of the AFM tip employed as a membrane indenter. Our study is focused on phase-separated fluid-gel lipid membranes at room temperature. In a small tip radius regime (≈ 2 nm) and in the case of fluid phase membranes, we show that the tip can penetrate through the membrane minimizing molecular vertical compression and in absence of molecular membrane rupture. In this case, AFM indentation experiments cannot assess the vertical membrane Young's modulus. In agreement with the data reported in the literature, in the case of larger indenters (>2 nm) SLBs can be compressed leading to an evaluation of Young's modulus and membrane maximal withstanding force before rupture. We show that such force increases with the indenter in agreement with the existing theoretical frame. Finally, we demonstrate that the latter has no influence on the number of molecules involved in the rupture process that is observed to be constant and rather dependent on the indenter chemical composition.

Micromechanical heterogeneity of the rat pia-arachnoid complex

To better understand the onset of damage occurring in the brain upon traumatic events, it is essential to analyze how external mechanical loads propagate through the skull and meninges and down to the brain cortex. However, despite their crucial role as structural dampers protecting the brain, the mechanical properties and dynamic behavior of the meningeal layers are still poorly understood. Here, we characterized the local mechanical heterogeneity of rat pia-arachnoid complex (PAC) at the microscale via atomic force microscopy (AFM) indentation experiments to understand how microstructural variations at the tissue level can differentially affect load propagation. By coupling AFM mechanical testing with fresh tissue immunofluorescent staining, we could directly observe the effect of specific anatomical features on the local mechanical properties of tissue. We observed a two-fold stiffening of vascularized tissue when compared to non-vascularized PAC (with instantaneous Young’s modulus distribution means of 1.32 ± 0.03 kPa and 2.79 ± 0.08 kPa, respectively), and statistically significant differences between regions of low- and high-vimentin density, reflecting trabecular density (with means of 0.67 ± 0.05 kPa and 1.29 ± 0.06 kPa, respectively). No significant differences were observed between cortical and cerebellar PAC. Additionally, by performing force relaxation experiments at the AFM, we identified the characteristic time constant τ1 of PAC tissue to be in the range of 2.7 ± 1.2 s to 3.1 ± 0.9 s for the different PAC regions analyzed. Taken together, the results presented point at the complex biomechanical nature of the meningeal tissue, and underscore the need to account for its heterogeneity when modeling its behavior into finite element simulations or other computational methods enabling the prediction of load propagation during injury events. Statement of SignificanceThe meningeal layers are pivotal in shielding the brain during injury events, yet the mechanical properties of this complex biological interface are still under investigation. Here, we present the first anatomically-informed micromechanical characterization of mammalian pia-arachnoid complex (PAC). We developed a protocol for the isolation and fresh immunostaining of rat PAC and subjected the tissue to AFM indentation and relaxation experiments, while visualizing the local anatomy via fluorescence microscopy. We found statistically significant variations in regional PAC stiffness according to the degree of vascularization and trabecular cell density, besides identifying the tissue’s characteristic relaxation constant. In essence, this study captures the relationship between anatomy and mechanical heterogeneity in a key component of the brain-skull interface for the first time.

Molecular dynamics simulation of cell membrane penetration by atomic force microscopy tip

In recent years, a delivery system based on atomic force microscopy (AFM) has been developed to transport nucleic acids, proteins and drugs to single adherent cell by controlling the indentation process. However, the transportation efficiency is limited by the low penetration rate of the common commercial AFM probe. The tip of commercial AFM probe is blunt and it is hard for blunt tip to insert into the soft cell membrane. In this study, dissipative particle dynamics (DPD) simulations were applied to investigate the penetration process of the AFM probe into the cell membrane subjected to different strain states. It was observed that the AFM tip moved down a shorter distance to penetrate the stretched lipid membrane compared with unstretched membrane. Moreover, the threshold value of penetrating force decreased as cell membrane extended. The short indentation time and small force can reduce the probability of cell membrane collapse, therefore it is easier for the AFM tip to penetrate the cell. We also performed the AFM indentation experiments via AFM to investigate the relationship between penetrating force and indentation speed. This work provides a potential way to improve the efficiency of cell transfection by using the AFM delivery system.

Dehomogenized Elastic Properties of Heterogeneous Layered Materials in AFM Indentation Experiments

Atomic force microscopy (AFM) is used to study mechanical properties of biological materials at submicron length scales. However, such samples are often structurally heterogeneous even at the local level, with different regions having distinct mechanical properties. Physical or chemical disruption can isolate individual structural elements but may alter the properties being measured. Therefore, to determine the micromechanical properties of intact heterogeneous multilayered samples indented by AFM, we propose the Hybrid Eshelby Decomposition (HED) analysis, which combines a modified homogenization theory and finite element modeling to extract layer-specific elastic moduli of composite structures from single indentations, utilizing knowledge of the component distribution to achieve solution uniqueness. Using finite element model-simulated indentation of layered samples with micron-scale thickness dimensions, biologically relevant elastic properties for incompressible soft tissues, and layer-specific heterogeneity of an order of magnitude or less, HED analysis recovered the prescribed modulus values typically within 10% error. Experimental validation using bilayer spin-coated polydimethylsiloxane samples also yielded self-consistent layer-specific modulus values whether arranged as stiff layer on soft substrate or soft layer on stiff substrate. We further examined a biophysical application by characterizing layer-specific microelastic properties of full-thickness mouse aortic wall tissue, demonstrating that the HED-extracted modulus of the tunica media was more than fivefold stiffer than the intima and not significantly different from direct indentation of exposed media tissue. Our results show that the elastic properties of surface and subsurface layers of microscale synthetic and biological samples can be simultaneously extracted from the composite material response to AFM indentation. HED analysis offers a robust approach to studying regional micromechanics of heterogeneous multilayered samples without destructively separating individual components before testing.

Are elastic moduli of biological cells depth dependent or not? Another explanation using a contact mechanics model with surface tension.

Atomic force microscopy (AFM) has become the most commonly used tool to measure the mechanical properties of biological cells. In AFM indentation experiments, the Hertz and Sneddon models of contact mechanics are usually adopted to extract the elastic modulus by analyzing the load-indent depth curves for spherical and conical tips, respectively. However, the effects of surface tension, neglected in existing contact models, become more significant in indentation responses due to the lower elastic moduli of living cells. Here, we present two simple yet robust relations between load and indent depth considering surface tension effects for spherical and conical indentations, through dimensional analysis and finite element simulations. When the indent depth is smaller than the intrinsic length defined as the ratio of surface tension to elastic modulus, the elastic modulus obtained by classical contact mechanics theories would be overestimated. Contrary to the majority of reported results, we find that the elastic modulus of a cell could be independent of indent depths if surface tension is taken into account. Our model seems to be in agreement with experimental data available. A comprehensive comparison will be done in the future.

Asymptotic approach to analyze the mechanical properties of biological cells

The Hertz model assuming a small indentation over the infinite hemisphere has been most widely applied to calculate elastic moduli of biological cells from data obtained by atomic force microscopy (AFM)-based indentation experiments. Previously reported experiments were mostly performed with a low stress to satisfy the Hertz assumption. In spite of its importance, mechanical heterogeneity observed in the high stress regime is often ignored due to the violation of the Hertz assumption. In this study, we have performed the hyperbolic fit modified from the Hertz model considering the asymptotic behaviors of layered structures in order to corroborate this issue. We demonstrate that our asymptotic approach confirmed the self-consistent elastic behavior of the cell cortex regardless of the applied stress regime. In addition, we have determined the elastic moduli of cellular regions beyond the cell cortex, where previous AFM indentation experiments could not easily access. We conclude that our asymptotic approach using hyperbolic fits provides a new empirically analytic mean to quantify the mechanical properties of biological cells including the high stress regime.

On the determination of elastic moduli of cells by AFM based indentation

The atomic force microscopy (AFM) has been widely used to measure the mechanical properties of biological cells through indentations. In most of existing studies, the cell is supposed to be linear elastic within the small strain regime when analyzing the AFM indentation data. However, in experimental situations, the roles of large deformation and surface tension of cells should be taken into consideration. Here, we use the neo-Hookean model to describe the hyperelastic behavior of cells and investigate the influence of surface tension through finite element simulations. At large deformation, a correction factor, depending on the geometric ratio of indenter radius to cell radius, is introduced to modify the force-indent depth relation of classical Hertzian model. Moreover, when the indent depth is comparable with an intrinsic length defined as the ratio of surface tension to elastic modulus, the surface tension evidently affects the indentation response, indicating an overestimation of elastic modulus by the Hertzian model. The dimensionless-analysis-based theoretical predictions, which include both large deformation and surface tension, are in good agreement with our finite element simulation data. This study provides a novel method to more accurately measure the mechanical properties of biological cells and soft materials in AFM indentation experiments.

Nanomechanical Investigation of Soft Biological Cell Adhesion using Atomic Force Microscopy

Mechanical coupling between living cells is a complex process that is important for a variety of biological processes. In this study the effects of specific biochemical treatment on cell-to-cell adhesion and single cell mechanics were systematically investigated using atomic force microscopy (AFM) single cell force spectroscopy. Functionalised AFM tipless cantilevers were used for attaching single suspended cells that were brought in contact with substrate cells. Cell-to-cell adhesion parameters, such as maximum unbinding force (F max) and work or energy of detachment (W D), were extracted from the retraction force–displacement (F–d) curves. AFM indentation experiments were performed by indenting single cells with a spherical microbead attached to the cantilever. Hertzian contact model was applied to determine the elastic modulus (E) of single cells. Following treatment of the cells with neutralising antibody for epithelial (E)-cadherin, F max was increased by 25%, whereas W D decreased by 11% in response to a 43% increase in E. The results suggest that although the adhesion force between cells was increased after treatment, the energy of adhesion was decreased due to the reduced displacement separation as manifested by the loss of elastic deformation. Conclusively, changes in single cell mechanics are important underlying factors contributing to cell-to-cell adhesion and hence cytomechanical characterization is critical for cell adhesion measurements.

A novel contact model for AFM indentation experiments on soft spherical cell-like particles.

The use of the simple Hertz model for the analysis of Atomic Force Microscopy (AFM) force-distance curves measured on soft spherical cell-like particles leads to significant underestimations of the objects Young's modulus E. To correct this error, a mixed double contact model (based on the simple Hertz model and the Johnson-Kendall-Roberts (JKR) model) was derived. The model considers two independent particle deformation sites: (i) the upper part of the particle is deformed by the AFM indenter, (ii) the bottom part is deformed by the substrate, which is usually unnoticed. It becomes apparent that for soft particles even small forces between substrate and particle can influence the resulting force-distance curves. For instance we show, that a gravity-induced compression on the particle bottom side can have significant influence on the measurements. To highlight these observations, the deviation of the particle Young's modulus E between the simple Hertz model and our model is calculated. This error strongly depends on the ratio of the three involved radii: (i) the radius of the AFM indenter, (ii) the radius of the particle and (iii) the radius of the substrate as well as on the acting gravity force. Overall, the analysis suggests that for nanoscopic indenters the deviation is negligible, whereas the use of microscopic indenters results in significant errors that can be corrected via the presented model. This is important especially for very soft particles, since larger indenters can achieve higher signal to noise ratios. Furthermore, the applicability of the model was confirmed by indentation experiments on hydrogel microbeads. The mixed double contact model is applicable to a large range of indenter geometries and can be adapted for other contact models.

Comparative study on the differential mechanical properties of human liver cancer and normal cells

Although cancerous cells and normal cells are known to have different elasticity values, there have been inconsistent reports in terms of the actual and relative values for these two cell types depending on the experimental conditions. This paper investigated the mechanical characterization of normal hepatocytes (THLE-2) and hepatocellular carcinoma cells (HepG2) using atomic force microscopy indentation experiments and the Hertz–Sneddon model, and the results were confirmed by an independent de-adhesion assay. To improve the reliability of the data, we considered the effects of tip geometry and indentation depth on the measured elasticity of the cells. This study demonstrated that THLE-2 cells had a higher elastic modulus compared with the HepG2 cells and that this difference was more significant when a conical tip was used. The inhibitor study indicated that this difference in the mechanical properties of THLE-2 and HepG2 cells was mainly attributed to differential arrangements in the cytoskeletal networks of actin filaments. An independent de-adhesion assay also confirmed that THLE-2 cells had a higher elastic modulus compared with the HepG2 cells, which resulted in a shorter time constant for cellular contractility.

Mechanical Tomography of Human Corneocytes with a Nanoneedle

Atomic force microscopy (AFM) can image biological samples and characterize their mechanical properties. However, the low aspect ratio of standard AFM probes typically limits these measurements to surface properties. Here, the intracellular mechanical behavior of human corneocytes is determined using "nanoneedle" AFM probes. The method evaluates the forces experienced by a nanoneedle as it is pushed into and then retracted from the cell. Indentation loops yield the stiffness profile and information on the elastic and nonelastic mechanical properties at a specific depth below the surface of the corneocytes. A clear difference between the softer ∼50-nm-thick external layer and the more rigid internal structure of corneocytes is apparent, which is consistent with the current understanding of the structure of these cells. There are also significant variations in the mechanical properties of corneocytes from different volunteers. The small diameter of the nanoneedle allows this "mechanical tomography" to be performed with high spatial resolution, potentially offering an opportunity to detect biomechanical changes in corneocytes because of, e.g., environmental factors, aging, or dermatological pathologies.

Method for quantitative measurements of the elastic modulus of biological cells in AFM indentation experiments

Here we overview and further develop a quantitative method to measure mechanics of biological cells in indentation experiments, which is based on the use of atomic force microscopy (AFM). We demonstrate how the elastic modulus of the cell body should be measured when the cellular brush is taken into account. The brush is an essential inelastic part of the cell, which surrounds all eukaryotic (the brush is mostly microvilli and glycocalyx) and gram-negative prokaryotic cells (the brush is polysaccharides). The other main feature of the described method is the use of a relatively dull AFM probe to stay in the linear stress–strain regime. In particular, we show that the elastic modulus (aka the Young’s modulus) of cells is independent of the indentation depth up to 10–20% deformation for the eukaryotic cells studied here. Besides the elastic modulus, the method presented allows obtaining the parameters of cellular brush, such as the effective length and grafting density of the brush. Although the method is demonstrated on eukaryotic cells, it is directly applicable for all types of cells, and even non-biological soft materials surrounded by either a brush or any field of long-range forces.

Strength by atomic force microscopy (AFM): Molecular dynamics of water layer squeezing on magnesium oxide

Localised strength testing of materials is often carried out in an atomic force microscope (AFM), as foreseen by Kelly in his book Strong Solids (Clarendon Press, Oxford, 1966). During AFM indentation experiments, contamination can strongly influence the observed strength and theoretical interpretation of the results is a major problem. Here, we use molecular dynamics computer modelling to describe the contact of NaCl and MgO crystal probes onto surfaces, comparable to an AFM experiment. Clean NaCl gave elastic, brittle behaviour in contact simulations at 300 K, whereas MgO was more plastic, leading to increased toughness. This paper also considers the strength of an oxide substrate contaminated by water molecules and tested by indentation with a pyramidal probe of oxide crystal. Recent theory on the effect of liquid contaminant layers on surface strength has been mainly focussed on Lennard Jones (LJ) molecules with some studies on alcohols and water, described by molecular dynamics, which allows the molecules to be squeezed out as the crystal lattice is deformed. In this work, we have focused on water by studying the forces between a magnesium oxide (MgO) atomic force microscope (AFM) probe and an MgO slab. Force versus separation has been plotted as the AFM probe was moved towards and away from the substrate. Simulation results showed that the water layers could be removed in steps, giving up to four force peaks. The last monolayer of water could not be squeezed out, even at pressures where MgO deformed plastically. Interestingly, with water present, strength was reduced, but more in tensile than compressive measurements. In conclusion, water contaminating the oxide surface in AFM strength testing is structured. Water layer squeezing removal can be predicted by molecular modelling, which may be verified by AFM experiments to show that water can influence the strength of perfect crystals at the nanometre scale.

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 degrees C agree well with the theoretical prediction and are used to determine the cell elasticity modulus and indenter-cell work of adhesion.

Adhesion and elasticity in nanoscale indentation

The present work proposes an extension of Oliver-Pharr analysis [J. Mater. Res. 7, 1564 (1992)] of unloading force-displacement data obtained in nanoscale indentation experiments to account for contact adhesion. The loading force is considered a sum of the contact elastic and adhesion forces. During the unloading, both forces suffer variations. For conical geometry, the unloading force-displacement curve is described by a sum of a quadratic term (accounting for the elastic force) and a linear term (accounting for adhesive force). Results of atomic force microscopy indentation experiments performed with sharpened silicon tips on poly(dimethylsiloxane) agreed well with the prediction of the proposed theoretical model.