Experimental validation of atomic force microscopy-based cell elasticity measurements

Atomic force microscopy (AFM) is widely used for measuring the elasticity of living cells yielding values ranging from 100 Pa to 100 kPa, much larger than those obtained using bead-tracking microrheology or micropipette aspiration (100-500 Pa). AFM elasticity measurements appear dependent on tip geometry with pyramidal tips yielding elasticities 2-3 fold larger than spherical tips, an effect generally attributed to the larger contact area of spherical tips. In AFM elasticity measurements, experimental force-indentation curves are analyzed using contact mechanics models that infer the tip-cell contact area from the tip geometry and indentation depth. The validity of these assumptions has never been verified. Here we utilize combined AFM-confocal microscopy of epithelial cells expressing a GFP-tagged membrane marker to directly characterize the indentation geometry and measure the indentation depth. Comparison with data derived from AFM force-indentation curves showed that the experimentally measured contact area for spherical tips agrees well with predicted values, whereas for pyramidal tips, the contact area can be grossly underestimated at forces larger than ∼0.2 nN leading to a greater than two-fold overestimation of elasticity. These data suggest that a re-examination of absolute cellular elasticities reported in the literature may be necessary and we suggest guidelines for avoiding elasticity measurement artefacts introduced by extraneous cantilever-cell contact.

Implications of a poroelastic cytoplasm for the dynamics of animal cell shape

Two views have dominated recent discussions of the physical basis of cell shape change during migration and division of animal cells: the cytoplasm can be modeled as a viscoelastic continuum, and the forces that change its shape are generated only by actin polymerization and actomyosin contractility in the cell cortex. Here, we question both views: we suggest that the cytoplasm is better described as poroelastic, and that hydrodynamic forces may be generally important for its shape dynamics. In the poroelastic view, the cytoplasm consists of a porous, elastic solid (cytoskeleton, organelles, ribosomes) penetrated by an interstitial fluid (cytosol) that moves through the pores in response to pressure gradients. If the pore size is small (30-60nm), as has been observed in some cells, pressure does not globally equilibrate on time and length scales relevant to cell motility. Pressure differences across the plasma membrane drive blebbing, and potentially other type of protrusive motility. In the poroelastic view, these pressures can be higher in one part of a cell than another, and can thus cause local shape change. Local pressure transients could be generated by actomyosin contractility, or by local activation of osmogenic ion transporters in the plasma membrane. We propose that local activation of Na(+)/H(+) antiporters (NHE1) at the front of migrating cells promotes local swelling there to help drive protrusive motility, acting in combination with actin polymerization. Local shrinking at the equator of dividing cells may similarly help drive invagination during cytokinesis, acting in combination with actomyosin contractility. Testing these hypotheses is not easy, as water is a difficult analyte to track, and will require a joint effort of the cytoskeleton and ion physiology communities.

A microstructural finite element simulation of mechanically induced bone formation

A finite element method to simulate the formation of an interconnected trabectular bone microstructure oriented with respect to applied in vivo mechanical forces is introduced and quantitatively compared to experimental data from a hydraulic bone chamber implant model. Randomly located 45 microm mineralized nodules were used as the initial condition for the model simulations to represent an early stage of intramembranous bone formation. Boundary conditions were applied consistent with the mechanical environment provided by the in vivo bone chamber model. A two-dimensional repair simulation algorithim that incorporated strain energy density (SED), SED gradient, principal strain, or principal strain gradient as the local objective criterion was utilized to simulate the formation of an oriented trabecular bone microstructure. The simulation solutions were convergent, unique, and relatively insensitive to the assumed initial distribution of mineralized nodules. Model predictions of trabecular bone morphology and anisotropy were quantitatively compared to experimental results. All simulations produced structures that qualitatively resembled oriented trabecular bone. However only simulations utilizing a gradient objective criterion yielded results quantitatively similar to in vivo observations. This simulation approach coupled with an experimental model that delivers controlled in vivo mechanical stimuli can be utilized to study the relationship between physical factors and microstructural adaptation during bone repair.

Atomic force microscopy can be used to mechanically stimulate osteoblasts and evaluate cellular strain distributions

In this study, atomic force microscopy (AFM) was used to mechanically stimulate primary osteoblasts. In response to mechanical force applied by the AFM, the indented cell increased its intracellular calcium concentration. The material properties of the cell could be estimated and the membrane strains calculated. We proceeded to validate this technique experimentally and a 20% error was found between the predicted and the measured diameter of indentation. We also determined the strain distributions within the cell that result from AFM indentation using a simple finite element model. This enabled us to formulate hypotheses as to the mechanism through which cells may sense the applied mechanical strains. Finally, we report the effect of the Poisson ratio and the cell thickness on the strain distributions. Varying the Poisson ratio did not change the order of magnitude of the strains; whereas the cellular thickness dramatically changed the order of magnitude of the cellular strains. We conclude that AFM can be used for controlled mechanical stimulation of osteoblasts and that cellular strain distributions can be computed with a good accuracy when the cell is indented in its highest part.

New technologies in scanning probe microscopy for studying molecular interactions in cells

Atomic force microscopy (AFM) is a specialised form of scanning probe microscopy, which was invented by Binnig and colleagues in 1986. Since then, AFM has been increasingly used to study biomedical problems. Because of its high resolution, AFM has been used to examine the topography or shape of surfaces, such as during the molecular imaging of proteins. This, combined with the ability to operate under known force regimes, makes AFM technology particularly useful for measuring intermolecular bond forces and assessing the mechanical properties of biological materials. Many of the constraints (e.g. complex instrumentation, slow acquisition speeds and poor vertical range) that previously limited the use of AFM in cell biology are now beginning to be resolved. Technological advances will enable AFM to challenge both confocal laser scanning microscopy and scanning electron microscopy as a method for carrying out three-dimensional imaging. Its use as both a precise micro-manipulator and a measurement tool will probably result in many novel and exciting applications in the future. In this article, we have reviewed some of the current biological applications of AFM, and illustrated these applications using studies of the cell biology of bone and integrin-mediated adhesion.

Adapting atomic force microscopy for cell biology

We present details of our AFM modifications to produce an adaptable imaging system for the cell biologist. We have designed and validated a new inverted microscope interface and a scan head with increased Z-range, based upon the TopoMetrix Explorer AFM. We have utilised these changes, together with home-made glass ball cantilevers, to obtain topographical information over cells with large Z-dimension (over 15 microm high), and mapped calcitonin-calcitonin receptor binding forces in living bone cells. We conclude that modified AFM can be used to evaluate intermolecular events in living cells and that this approach will ensure general application to the study of receptor-ligand interactions under truly physiological conditions.

Improving the local solution accuracy of large-scale digital image-based finite element analyses

Digital image-based finite element modeling (DIBFEM) has become a widely utilized approach for efficiently meshing complex biological structures such as trabecular bone. While DIBFEM can provide accurate predictions of apparent mechanical properties, its application to simulate local phenomena such as tissue failure or adaptation has been limited by high local solution errors at digital model boundaries. Furthermore, refinement of digital meshes does not necessarily reduce local maximum errors. The purpose of this study was to evaluate the potential to reduce local mean and maximum solution errors in digital meshes using a post-processing filtration method. The effectiveness of a three-dimensional, boundary-specific filtering algorithm was found to be mesh size dependent. Mean absolute and maximum errors were reduced for meshes with more than five elements through the diameter of a cantilever beam considered representative of a single trabecula. Furthermore, mesh refinement consistently decreased errors for filtered solutions but not necessarily for non-filtered solutions. Models with more than five elements through the beam diameter yielded absolute mean errors of less than 15% for both Von Mises stress and maximum principal strain. When applied to a high-resolution model of trabecular bone microstructure, boundary filtering produced a more continuous solution distribution and reduced the predicted maximum stress by 30%. Boundary-specific filtering provides a simple means of improving local solution accuracy while retaining the model generation and numerical storage efficiency of the DIBFEM technique.

The accuracy of digital image-based finite element models

Digital image-based finite element meshing is an alternative approach to time-consuming conventional meshing techniques for generating realistic three-dimensional (3D) models of complex structures. Although not limited to biological applications, digital image-based modeling has been used to generate structure-specific (i.e., non-generic) models of whole bones and trabecular bone microstructures. However, questions remain regarding the solution accuracy provided by the digital meshing approach, particularly at model or material boundaries. The purpose of this study was to compare the accuracy of digital and conventional smooth boundary models based on theoretical solutions for a two-dimensional (2D) compression plate and a 3D circular cantilever beam. For both the plate and beam analyses, the predicted solution at digital model boundaries was characterized by local oscillations, which produced potentially high errors within individual boundary elements. Significantly, however, the digital model boundary solution oscillated approximately about the theoretical solution. A marked improvement in solution accuracy was therefore achieved by considering average results within a region composed of several elements. Absolute errors for Von Mises stress averaged over the beam cross section, for example, converged to less than 4 percent, and the predicted free-end displacement of the cantilever beam was within 1 percent of the theoretical solution. Analyses at several beam orientations and mesh resolutions suggested a minimum discretization of three to four digital finite elements through the beam cross section to avoid high numerical stiffening errors under bending.