Atomistic uniaxial tension tests: investigating various many-body potentials for their ability to produce accurate stress strain curves using molecular dynamics simulations

Molecular dynamics simulations, which take place on the atomistic scale, are now being used to predict the influence of atomistic processes on macro-scale mechanical properties. However, there is a lack of clear understanding on which potential should be used when attempting to obtain these properties. Moreover, many MD studies that do test mechanical properties do not actually simulate the macro-scale laboratory tension tests used to obtain them. As such, the purpose of the current study was to evaluate the various types of potentials for their accuracy in predicting the mechanical properties of iron from an atomistic uniaxial tension test at room temperature. Results demonstrated that while EAM and MEAM potentials all under predicted the elastic modulus at room temperature, the Tersoff and ReaxFF potentials were significantly more accurate. Unlike EAM and MEAM, both the Tersoff and ReaxFF potentials are bond order based. Therefore, these results demonstrate the importance of considering bonding between atoms when modelling tensile tests. In addition, the ReaxFF potential also accurately predicted the Poisson's ratio, allowing for complete characterisation of the material's behaviour. Overall, these findings highlight the need to understand the capabilities and limitations of each potential before application to a problem outside of the initial intended use.     

Evaluation of Stem Cell-to-Tenocyte Differentiation By Atomic Force Microscopy to Measure Cellular Elastic Moduli

In the present study, we evaluated whether stem cell-to-tenocyte differentiation could be evaluated via measurement of the mechanical properties of the cell. We used mechanical uniaxial cyclic stretching to induce the differentiation of human bone marrow mesenchymal stem cells into tenocytes. The cells were subjected to cyclic elongation of 10 or 15 % at a cyclic frequency of 1 Hz for 24 or 48 h, and differentiation was assessed by real-time PCR (rtPCR) determination of messenger RNA expression levels for four commonly used markers of stem cell-to-tenocyte differentiation: type I collagen, type III collagen, tenascin-C, and scleraxis. The rtPCR results showed that cells subjected to 10 % cyclic elongation for 24 or 48 h differentiated into tenocytes. Atomic force microscopy (AFM) was then used to measure the force curves around the cell nuclei, and the AFM data were used to calculate the elastic moduli of the cell surfaces. The elastic modulus values of the control (non-stretched) cells differed significantly from those of cells stretched at 10 % for 24 or 48 h (P < 0.01). Confocal fluorescence microscopic observations of actin stress fibers suggested that the change in elastic modulus was ascribable to the development of the cellular cytoskeleton during the differentiation process. Therefore, we conclude that the atomic force microscopic measurement of the elastic modulus of the cell surface can be used to evaluate stem cell-to-tenocyte differentiation.     

The role of an effective isotropic tissue modulus in the elastic properties of cancellous bone.

Conceptually, the elastic characteristics of cancellous bone could be predicted directly from the trabecular morphology--or architecture--and by the elastic properties of the tissue itself. Although hardly any experimental evidence exists, it is often implicitly assumed that tissue anisotropy has a negligible effect on the apparent elastic properties of cancellous bone. The question addressed in this paper is whether this is actually true. If it is, then micromechanical finite element analysis (micro-FEA) models, representing trabecular architecture, using an 'effective isotropic tissue modulus' should be able to predict apparent elastic properties of cancellous bone. To test this, accurate multi-axial compressive mechanical tests of 29 whale bone specimens were simulated with specimen-specific micro-FEA computer models built from true three-dimensional reconstructions. By scaling the micro-FEA predictions by a constant tissue modulus, 92% of the variation of Young's moduli determined experimentally could be explained. The correlation even increased to 95% when the micro-FEA moduli were scaled to the isotropic tissue moduli of individual specimens. Excellent agreement was also found in the elastic symmetry axes and anisotropy ratios. The prediction of Poisson's ratios was somewhat less precise at 85% correlation. The results support the hypothesis; for practical purposes, the concept of an 'effective isotropic tissue modulus' concept is a viable one. They also suggest that the value of such a modulus for individual cases might be inferred from the average tissue density, hence the degree of mineralization. Future studies must clarify how specific the tissue modulus should be for different types of bone if adequate predictions of elastic behavior are to be made in this way.     

Biomechanics of nodal, seminal and lateral roots of barley: effects of diameter, waterlogging and mechanical impedance

Background and aims

Biomechanical properties of cereal root systems largely control both resistance to root lodging and their ability to stabilise soil. Abiotic stresses can greatly modify root system growth and form. In this paper the effect of waterlogging and moderate mechanical impedance on root biomechanics is studied for both lateral roots and the main axes of barley.

Methods

Barley (Hordeum vulgare) plants were subjected to transient water-logging and moderate mechanical impedance in repacked soil columns. Roots were excavated, separated into types (nodal, seminal or lateral) and tested in tension to measure strength and elastic modulus.

Results

Water-logging and mechanical impedance substantially changed root system growth whilst root biomechanical properties were affected by waterlogging. Root strength was generally greater in thin roots and depended on root type. For example, seminal roots 0.4–0.6 mm in diameter were approximately seven times stronger and five times stiffer than lateral roots of the same diameter when mechanically impeded. Root sample populations typically exhibited negative power-law relationships between root strength and diameter for all root types. Mechanical impedance slowed seminal root elongation by approximately 50 % and resulted in a 15 % and 11 % increase in the diameter of in nodal and seminal roots respectively. Power-law relationships between root diameter and root biomechanical properties corresponded to the different root types. Coefficients for between root diameter, strength and elastic modulus improved when separated by root type, with R² values increasing in some roots from 0.05 to 0.71 for root strength and 0.08 to 0.74 for elastic modulus.

Conclusions

Moderate mechanical impedance did not influence the tensile strength of roots, but, waterlogging diminished the relationship between root strength and diameter. Separation of root type improved predictions of root strength and elastic modulus using power-law regressions.     

REACH coarse-grained biomolecular simulation: transferability between different protein structural classes

Coarse graining of protein interactions provides a means of simulating large biological systems. The REACH (Realistic Extension Algorithm via Covariance Hessian) coarse-graining method, in which the force constants of a residue-scale elastic network model are calculated from the variance-covariance matrix obtained from atomistic molecular dynamics (MD) simulation, involves direct mapping between scales without the need for iterative optimization. Here, the transferability of the REACH force field is examined between protein molecules of different structural classes. As test cases, myoglobin (all α), plastocyanin (all β), and dihydrofolate reductase (α/β) are taken. The force constants derived are found to be closely similar in all three proteins. An MD version of REACH is presented, and low-temperature coarse-grained (CG) REACH MD simulations of the three proteins are compared with atomistic MD results. The mean-square fluctuations of the atomistic MD are well reproduced by the CGMD. Model functions for the CG interactions, derived by averaging over the three proteins, are also shown to produce fluctuations in good agreement with the atomistic MD. The results indicate that, similarly to the use of atomistic force fields, it is now possible to use a single, generic REACH force field for all protein studies, without having first to derive parameters from atomistic MD simulation for each individual system studied. The REACH method is thus likely to be a reliable way of determining spatiotemporal motion of a variety of proteins without the need for expensive computation of long atomistic MD simulations.     

Stochastic multi-scale prediction on the apparent elastic moduli of trabecular bone considering uncertainties of biological apatite (BAp) crystallite orientation and image-based modelling

An assessment of the mechanical properties of trabecular bone is important in determining the fracture risk of human bones. Many uncertainty factors contribute to the dispersion of the estimated mechanical properties of trabecular bone. This study was undertaken in order to propose a computational scheme that will be able to predict the effective apparent elastic moduli of trabecular bone considering the uncertainties that are primarily caused by image-based modelling and trabecular stiffness orientation. The effect of image-based modelling which focused on the connectivity was also investigated. A stochastic multi-scale method using a first-order perturbation-based and asymptotic homogenisation theory was applied to formulate the stochastically apparent elastic properties of trabecular bone. The effective apparent elastic modulus was predicted with the introduction of a coefficient factor to represent the variation of bone characteristics due to inter-individual differences. The mean value of the predicted effective apparent Young's modulus in principal axis was found at approximately 460 MPa for respective 15.24% of bone volume fraction, and this is in good agreement with other experimental results. The proposed method may provide a reference for the reliable evaluation of the prediction of the apparent elastic properties of trabecular bone.     

REACH Coarse-Grained Simulation of a Cellulose Fiber

A molecular level understanding of the structure, dynamics and mechanics of cellulose fibers can aid in understanding the recalcitrance of biomass to hydrolysis in cellulosic biofuel production. Here, a residue-scale REACH (Realistic Extension Algorithm via Covariance Hessian) coarse-grained force field was derived from all-atom molecular dynamics (MD) simulations of the crystalline Iβ cellulose fibril. REACH maps the atomistic covariance matrix onto coarse-grained elastic force constants. The REACH force field was found to reproduce the positional fluctuations and low-frequency vibrational spectra from the all-atom model, allowing elastic properties of the cellulose fibril to be characterized using the coarse-grained force field with a speedup of >20 relative to atomistic MD on systems of the same size. The calculated longitudinal/transversal Young's modulus and the velocity of sound are in agreement with experiment. The persistence length of a 36-chain cellulose microcrystal was estimated to be ～380 μm. Finally, the normal-mode analysis with the REACH force field suggests that intrinsic dynamics might facilitate the deconstruction of the cellulose fibril from the hydrophobic surface.     

Progress in molecular modelling of DNA materials

The unique molecular recognition properties of DNA molecule, which store genetic information in cells, are responsible for the rise of DNA nanotechnology. In this article, we review the recent advances in atomistic and coarse-grained force fields along with simulations of DNA-based materials, as applied to DNA–nanoparticle assemblies for controlled material morphology, DNA–surface interactions for biosensor development and DNA origami. Evidently, currently available atomistic and coarse-grained representations of DNA are now at the stage of successfully reproducing and explaining experimentally observed phenomena. However, there is a clear need for the development of atomistic force fields which are robust at long timescales and in the improvement of the coarse-grained models.     

Measurement of the circumferential mechanical properties of the umbilical vein: experimental and numerical analyses

Coronary artery disease is responsible for almost 30% of all deaths worldwide. The saphenous vein and umbilical vein (UV) are the most common veins using for treatment as a coronary artery bypass graft (CABG). The mechanical properties of UV belonging to its long-term patency for CABG are very important. However, there is a lack of knowledge on the linear elastic and nonlinear hyperelastic mechanical properties of the UV. In this study, three stress definitions (second Piola–Kichhoff stress, engineering stress and true stress) and four strain definitions (Almansi–Hamel strain, Green–St Venant strain, engineering strain and true strain) are used to determine the elastic modulus, maximum stress and strain of eight human UVs under circumferential loading. The nonlinear mechanical behaviour of the UV is computationally investigated using Mooney–Rivlin hyperelastic model. A numerical finite element analysis is also carried out to simulate the constitutive modelling versus its numerical results. The results show that the Almansi–Hamel strain definition overestimates the elastic modulus while Green–St Venant strain definition underestimates the elastic modulus at different stress definitions. The true stress–true strain definition, which gives more accurate measurements of the tissue's response using the instantaneous values, reveals the Young's modulus and maximum stress of 2.18 and 6.01 MPa, respectively. The Mooney–Rivlin material model is well represented by the nonlinear mechanical behaviour of the UV. The findings of this study could have implications not only for understanding the extension and rupture mechanism of UV but also for interventions and surgeries, including balloon angioplasty, bypass and stenting.     

Macrophage adhesion on fibronectin evokes an increase in the elastic property of the cell membrane and cytoskeleton: an atomic force microscopy study

Interactions between cells and microenvironments are essential to cellular functions such as survival, exocytosis and differentiation. Cell adhesion to the extracellular matrix (ECM) evokes a variety of biophysical changes in cellular organization, including modification of the cytoskeleton and plasma membrane. In fact, the cytoskeleton and plasma membrane are structures that mediate adherent contacts with the ECM; therefore, they are closely correlated. Considering that the mechanical properties of the cell could be affected by cell adhesion-induced changes in the cytoskeleton, the purpose of this study was to investigate the influence of the ECM on the elastic properties of fixed macrophage cells using atomic force microscopy. The results showed that there was an increase (~50 %) in the Young’s modulus of macrophages adhered to an ECM-coated substrate as compared with an uncoated glass substrate. In addition, cytochalasin D-treated cells had a 1.8-fold reduction of the Young’s modulus of the cells, indicating the contribution of the actin cytoskeleton to the elastic properties of the cell. Our findings show that cell adhesion influences the mechanical properties of the plasma membrane, providing new information toward understanding the influence of the ECM on elastic alterations of macrophage cell membranes.     

A simple Hill element-nonlinear spring model of muscle contraction biomechanics

The purpose of this study was to develop a model to predict the mechanical response of muscles during isometric tetanic, afterloaded isotonic and isovelocity shortening contractions. Two versions of the model were developed. Both incorporated a contractile element that obeyed a Hill force-velocity relationship and a series elastic element. In a quadratic spring version, the series elastic element force was represented as proportional to the square of the stretch; in a cubic spring version, it was represented as proportional to the cube of the stretch. Both versions provided closed-form equations for response predictions that involved four independent parameters. Once the four parameters were chosen, each of these responses could be predicted. Model validity was established by comparing predicted and observed responses in slow and fast hindlimb muscles of rodents. Significant model-predicted responses seldom differed by more than 15% from experimental values. The model can provide insights into how changes in individual properties affect the overall mechanical behavior of muscles in a variety of circumstances and reduce the need for collection of experimental data.     

Validation of a voxel-based FE method for prediction of the uniaxial apparent modulus of human trabecular bone using macroscopic mechanical tests and nanoindentation

Assessment of the mechanical properties of trabecular bone is of major biological and clinical importance for the investigation of bone diseases, fractures and their treatments. Finite element (FE) methods are getting increasingly popular for quantifying the elastic and failure properties of trabecular bone. In particular, voxel-based FE methods have been previously used to calculate the effective elastic properties of trabecular microstructures. However, in most studies, bone tissue moduli were assumed or back-calculated to match the apparent elastic moduli from experiments, which often lead to surprisingly low values when compared to nanoindentation results. In this study, voxel-based FE analysis of trabecular bone is combined with physical measures of volume fraction, micro-CT (microCT) reconstructions, uniaxial mechanical tests and specimen-specific nanoindentation tests for proper validation of the method. Cylindrical specimens of cancellous bone were extracted from human femurs and their volume fraction determined with Archimede's method. Uniaxial apparent modulus of the specimens was measured with an improved tension-compression testing protocol that minimizes boundary artefacts. Their microCT reconstructions were segmented to match the measured bone volume fraction and used to create full-size voxel models with 30-45 microm element size. For each specimen, linear isotropic elastic material properties were defined based on specific nanoindentation measurements of its embedded bone tissue. Linear FE analyses were finally performed to simulate the uniaxial mechanical tests. Additional parametric analyses were performed to evaluate the potential errors on the predicted apparent modulus arising from variations in segmentation threshold, tissue modulus, and the use of 125-mm(3) cubic sub-regions. The results demonstrate an excellent correspondence between experimental measures and FE predictions of uniaxial apparent modulus. In conclusion, the adopted voxel-based FE approach is found to be a robust method to predict the linear elastic properties of human cancellous bone, provided segmentation of the microCT reconstructions is carefully calibrated, tissue modulus is known a priori and the entire region of interest is included in the analysis.     

Effect of Matrix on Cardiomyocyte Viscoelastic Properties in 2D Culture

Cardiomyocyte phenotype changes significantly in 2D culture systems depending on the substrate composition and organization. Given the variety of substrates that are used both for basic cardiac cell culture studies and for regenerative medicine applications, there is a critical need to understand how the different matrices influence cardiac cell mechanics. In the current study, the mechanical properties of neonatal rat cardiomyocytes cultured in a subconfluent layer upon aligned and unaligned collagen and fibronectin matrices were assessed over a two week period using atomic force microscopy. The elastic modulus was estimated by fitting the Hertz model to force curve data and the percent relaxation was determined from stress relaxation curves. The Quasilinear Viscoelastic (QLV) and Standard Linear Solid (SLS) models were fit to the stress relaxation data. Cardiomyocyte cellular mechanical properties were found to be highly dependent on matrix composition and organization as well as time in culture. It was observed that the cells stiffened and relaxed less over the first 3 to 5 days in culture before reaching a plateau in their mechanical properties. After day 5, cells on aligned matrices were stiffer than cells on unaligned matrices and cells on fibronectin matrices were stiffer than cells on collagen matrices. No such significant trends in percent relaxation measurements were observed but the QLV model fit the data very well. These results were correlated with observed changes in cellular structure associated with culture on the different substrates and analyzed for cell-to-cell variability.     

Mechanical properties of hybridoma cells in batch culture

Summary Direct measurements throughout a batch culture of bursting force, bursting membrane tension, elastic area compressibility modulus, and the size of single hybridoma cells have been made by a novel micromanipulation technique. It has been found that the bursting membrane tension and compressibility modulus rise significantly in the rapid growth phase, and fall in the death phase. An approach is suggested for relating these mechanical properties to the shear sensitivity of the cells when they are exposed to shear stresses in flow fields. It is shown that reports of changes in hybridoma fragility during batch cultures, as measured using viscometers, might be explained using more fundamental micromanipulation measurements.     

Temperature and configurational effects on the Young’s modulus of poly (methyl methacrylate): a molecular dynamics study comparing the DREIDING,AMBER and OPLS force fields

The effects of the configuration and temperature on the Young’s modulus of poly (methyl methacrylate) (PMMA) have been studied using molecular dynamics simulations. For the DREIDING force field under ambient temperatures, increasing the number of monomers significantly increases the modulus of isotactic and syndiotactic PMMA while the isotactic form has a greater modulus. The effects of temperature on the modulus of isotactic PMMA have been simulated using the DREIDING, AMBER, and OPLS force fields. All these force fields predict the effects of temperature on the modulus from 200 to 350 K that are in close agreement with experimental values, while at higher temperatures the moduli are greater than those measured. The glass transition temperature determined by the force fields, based on the variation of the modulus with temperature, is greater than the experimental values, but when obtained from a plot of the volume as a function of the temperature, there is closer agreement. The Young’s moduli calculated in this study are in closer agreement to the experimental data than those reported by previous simulations.     

Analytical methods to determine the effective mesoscopic and macroscopic elastic properties of cortical bone

We compare theoretical predictions of the effective elastic moduli of cortical bone at both the meso- and macroscales. We consider the efficacy of three alternative approaches: the method of asymptotic homogenization, the Mori-Tanaka scheme and the Hashin-Rosen bounds. The methods concur for specific engineering moduli such as the axial Young's modulus but can vary for others. In a past study, the effect of porosity alone on mesoscopic properties of cortical bone was considered, taking the matrix to be isotropic. Here, we consider the additional influence of the transverse isotropy of the matrix. We make the point that micromechanical approaches can be used in two alternative ways to predict either the macroscopic (size of cortical bone sample) or mesoscopic (in between micro- and macroscales) effective moduli, depending upon the choice of representative volume element size. It is widely accepted that the mesoscale behaviour is an important aspect of the mechanical behaviour of bone but models incorporating its effect have started to appear only relatively recently. Before this only macroscopic behaviour was addressed. Comparisons are drawn with experimental data and simulations from the literature for macroscale predictions with particularly good agreement in the case of dry bone. Finally, we show how predictions of the effective mesoscopic elastic moduli can be made which retain dependence on the well-known porosity gradient across the thickness of cortical bone.     

Determination of mechanical properties of soft tissue scaffolds by atomic force microscopy nanoindentation

While the determination of mechanical properties of a hard scaffold is relatively straightforward, the mechanical testing of a soft tissue scaffold poses significant challenges due in part to its fragility. Here, we report a new approach for characterizing the stiffness and elastic modulus of a soft scaffold through atomic force microscopy (AFM) nanoindentation. Using collagen-chitosan hydrogel scaffolds as model soft tissue scaffolds, we demonstrated the feasibility of using AFM nanoindentation to determine a force curve of a soft tissue scaffold. A mathematical model was developed to ascertain the stiffness and elastic modulus of a scaffold from its force curve obtained under different conditions. The elastic modulus of a collagen-chitosan (80%/20%, v/v) scaffold is found to be 3.69 kPa. The scaffold becomes stiffer if it contains more chitosan. The elastic modulus of a scaffold composed of 70% collagen and 30% chitosan is about 11.6 kPa. Furthermore, the stiffness of the scaffold is found to be altered significantly by extracellular matrix deposited from cells that are grown inside the scaffold. The elastic modulus of collagen-chitosan scaffolds increased from 10.5 kPa on day 3 to 63.4 kPa on day 10 when human foreskin fibroblast cells grew inside the scaffolds. Data acquired from these measurements will offer new insights into understanding cell fate regulation induced by physiochemical cues of tissue scaffolds.     

Effect of Age and Cytoskeletal Elements on the Indentation-Dependent Mechanical Properties of Chondrocytes

Articular cartilage chondrocytes are responsible for the synthesis, maintenance, and turnover of the extracellular matrix, metabolic processes that contribute to the mechanical properties of these cells. Here, we systematically evaluated the effect of age and cytoskeletal disruptors on the mechanical properties of chondrocytes as a function of deformation. We quantified the indentation-dependent mechanical properties of chondrocytes isolated from neonatal (1-day), adult (5-year) and geriatric (12-year) bovine knees using atomic force microscopy (AFM). We also measured the contribution of the actin and intermediate filaments to the indentation-dependent mechanical properties of chondrocytes. By integrating AFM with confocal fluorescent microscopy, we monitored cytoskeletal and biomechanical deformation in transgenic cells (GFP-vimentin and mCherry-actin) under compression. We found that the elastic modulus of chondrocytes in all age groups decreased with increased indentation (15–2000 nm). The elastic modulus of adult chondrocytes was significantly greater than neonatal cells at indentations greater than 500 nm. Viscoelastic moduli (instantaneous and equilibrium) were comparable in all age groups examined; however, the intrinsic viscosity was lower in geriatric chondrocytes than neonatal. Disrupting the actin or the intermediate filament structures altered the mechanical properties of chondrocytes by decreasing the elastic modulus and viscoelastic properties, resulting in a dramatic loss of indentation-dependent response with treatment. Actin and vimentin cytoskeletal structures were monitored using confocal fluorescent microscopy in transgenic cells treated with disruptors, and both treatments had a profound disruptive effect on the actin filaments. Here we show that disrupting the structure of intermediate filaments indirectly altered the configuration of the actin cytoskeleton. These findings underscore the importance of the cytoskeletal elements in the overall mechanical response of chondrocytes, indicating that intermediate filament integrity is key to the non-linear elastic properties of chondrocytes. This study improves our understanding of the mechanical properties of articular cartilage at the single cell level.     

Primary Changes of the Mechanical Properties of Southern Bean Mosaic Virus upon Calcium Removal

The mechanical properties of viral shells are crucial determinates for the pathway and mechanism by which the genetic material leaves the capsid during infection and have therefore been studied by atomic force microscopy as well as by atomistic simulations. The mechanical response to forces from inside the capsid are found to be relevant, especially after ion removal from the shell structure, which is generally assumed to be essential during viral infection; however, atomic force microscopy measurements are restricted to probing the capsids from outside, and the primary effect of ion removal is still inaccessible. To bridge this gap, we performed atomistic force-probe molecular dynamics simulations of the complete solvated icosahedral shell of Southern Bean Mosaic Virus and compared the distribution of elastic constants and yielding forces on the icosahedral shell for probing from inside with the distribution of outside mechanical properties obtained previously. Further, the primary effect of calcium removal on the mechanical properties on both sides, as well as on their spatial distribution, is quantified. Marked differences are seen particularly at the pentamer centers, although only small structural changes occur on the short timescales of the simulation. This unexpected primary effect, hence, precedes subsequent effects due to capsid swelling. In particular, assuming that genome release is preceded by an opening of capsomers instead of a complete capsid bursting, our observed weakening along the fivefold symmetry axes let us suggest pentamers as possible exit ports for RNA release.     

Combining AFM and Acoustic Probes to Reveal Changes in the Elastic Stiffness Tensor of Living Cells

Knowledge of how the elastic stiffness of a cell affects its communication with its environment is of fundamental importance for the understanding of tissue integrity in health and disease. For stiffness measurements, it has been customary to quote a single parameter quantity, e.g., Young’s modulus, rather than the minimum of two terms of the stiffness tensor required by elasticity theory. In this study, we use two independent methods (acoustic microscopy and atomic force microscopy nanoindentation) to characterize the elastic properties of a cell and thus determine two independent elastic constants. This allows us to explore in detail how the mechanical properties of cells change in response to signaling pathways that are known to regulate the cell’s cytoskeleton. In particular, we demonstrate that altering the tensioning of actin filaments in NIH3T3 cells has a strong influence on the cell's shear modulus but leaves its bulk modulus unchanged. In contrast, altering the polymerization state of actin filaments influences bulk and shear modulus in a similar manner. In addition, we can use the data to directly determine the Poisson ratio of a cell and show that in all cases studied, it is less than, but very close to, 0.5 in value.