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

Abstract

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.

Introduction

In multicellular organisms, cells interact mechanically with the local environment, and studying their response to external stresses and strains is important for understanding their behavior in health and disease. Cells are dynamic structures and react to external mechanical stimuli that determine cell fate (1,2). Cells also react by changing their own internal architecture, which in turn leads to the alteration of mechanical properties of the cells themselves. For example, it has been shown that breast tumorigenesis is accompanied by extracellular matrix stiffening (3) and that cell stiffness can be a signature for tumors (4).

The mechanical behavior of a cell is governed by its complex internal structure, of which the cytoskeleton of filamentous actin (F-actin), intermediate filaments, and microtubules is believed to be the most important. This is in turn controlled by filament cross-linking, motor, and regulatory proteins. In the literature, as reviewed by both Lim et al. (5) and Kasza et al. (6), there are two alternative approaches that can be used to predict the mechanical behavior of cells: a bottom-up approach, in which the behavior of a cell is modeled as a network of polymer fibers, and a top-down approach, in which a cell is considered and modeled as a discrete entity. The bottom-up approach applies the principles of polymer physics to describe the behavior of filamentary networks (7–9) and as such may give greater insight into the mechanisms that generate the mechanical properties of cells. However, to make useful numerical predictions from these models, it is normal to describe the predicted behavior by using a continuum formulation such as elasticity or viscoelasticity. The top-down approach, normally based on data from whole-cell deformation experiments, seeks to define the behavior of a cell by directly determining a constitutive law that relates the action of external forces to local displacements and thus to formulate empirically the mechanical behavior within a preselected continuum formulation. Although the top-down approach lacks a direct physical interpretation, the continuum formulation can be used with powerful computational tools such as the finite-element method to produce predictions of cell behavior such as cell contractility and migration (10,11). Thus, although there are two distinct approaches to analyzing or predicting the mechanical response of a cell, the end result in both cases is the derivation of a relationship between mechanical force and displacement that simplifies the internal structure of a cell to that of an effective medium or continuum. We note that more sophisticated models of cells that mimic a variation in mechanical properties within a single cell still use a continuum formulation, but with the continuum showing local spatial variation in its properties (10,12).

The simplest constitutive law that can be used to model cell behavior is elastic behavior, where the deformation is proportional to the applied stress and is reversed once stress is removed. Time-dependent deformation of a cell can be modeled using more complex constitutive laws, two examples of which are given below.

• 1.Viscoelastic behavior has been used to model the behavior of cells (13). This is often used to model the behavior of solid polymers where deformation can occur by the relative displacement of polymer chains. Lim noted that an elastic model of a cell can be related to time-dependent viscoelastic models through the correspondence principle (5).
• 2.Poroelastic (or biphasic) behavior shows a time dependency controlled by the flow of liquid through nanoscale pores and channels in response to local differences in pressure and the deformation rate limited by its viscosity. Poroelasticity is used to model the deformation of some soft tissues (14) and, it has been suggested, may be applicable to the deformation of cells, given the large amount of fluid present in the cytoplasm (15).

However, it is difficult to distinguish conclusively between poroeleastic behavior and viscoelastic behavior without probes for fluid migration during cell deformation. A number of reviews of cell mechanical behavior state that over reasonably short timescales, of a few seconds or less, cell behavior can be adequately described by an elastic model (5,6,12). Thus, describing cell behavior using a simple elastic constitutive law is appropriate and will be used in the rest of this work.

In the literature, cell stiffness is commonly described by a single parameter, typically Young’s modulus or the shear modulus. However, cells are 3-dimensional (3D) objects, and in organisms, most cell types are normally embedded in an extracellular matrix. Hence, their resistance to deformation must be described in 3D Cartesian space. From simple deformation theory, it can be shown that in a homogeneous and isotropic medium, an arbitrary 3D elastic deformation consists of two independent components, a change in volume and a change in shape or shear (16). Thus, to describe mechanical responses to forces in 3D requires a minimum of two independent elastic constants, one to represent the resistance to a change in shape and the other its resistance to a change in volume.

Cell elastic properties

Structural rigidity is believed to be controlled by the cell cytoskeleton, which consists of three interconnected networks of polymer molecules comprising actin filaments, microtubules, and intermediate filaments. Cells can regulate intracellular tension and cell stiffness through actomyosin contractility (17,18); the actomyosin machinery is linked via adaptor proteins to transmembrane adhesion receptors, the integrins, which bind components of the extracellular matrix proteins. Microtubules are thought to resist the compression load formed by actomyosin tension (19). Chemicals that interfere with the structural polymers of the cytoskeleton are known to strongly influence cell mechanical properties. The actin filament network can be disrupted through the action of cytochalasin D or latrunculin B, hereby reducing the stiffness of cells (20). Conversely, microtubule-disrupting drugs, e.g., nocodazole, are associated with an increase in actin polymerization and bundling (21).

In this study, we use a top-down approach and assume that the properties of the cell can be adequately modeled using a constitutive law that describes elastic behavior. We assume that the mechanical properties of the cell can be modeled as an isotropic elastic continuum described by a symmetrical fourth-rank tensor, represented in matrix notation by (22)

$$\left(\begin{array}{cccccc}{C}_{11}& C_{12}& C_{12}& 0& 0& 0\\ -& C_{11}& C_{12}& 0& 0& 0\\ -& -& C_{11}& 0& 0& 0\\ -& -& -& C_{44}& 0& 0\\ -& -& -& -& C_{44}& 0\\ -& -& -& -& -& C_{44}\end{array}\right).$$ (1)

For an isotropic material, $C_{44}=\left(C_{11}-C_{12}\right)/2$; hence, there are two independent elastic constants (16).

For ease of interpretation, it is often more convenient to describe the elastic properties in terms of two of four conventional (or engineering) elastic constants, Young’s modulus, E; Poisson’s ratio, ν; the shear modulus, G; or the bulk modulus, K, where

$$E=\frac{\left(C_{11}-C_{12}\right)\left(C_{11}+2C_{12}\right)}{\left(C_{11}+C_{12}\right)}$$ (2a)

$$\nu =\frac{C_{12}}{\left(C_{11}+C_{12}\right)}$$ (2b)

$$G=\frac{\left(C_{11}-C_{12}\right)}{2}$$ (2c)

$$K=\frac{\left(C_{11}+2C_{12}\right)}{3}.$$ (2d)

Conversely, it is possible to fully describe the elastic properties of a cell by measuring any two of the above elastic constants.

In practice, most measurements of the elastic properties of cells are made using a single experimental technique to access one elastic constant. To get around this, Poisson’s ratio is usually assumed to be 0.5, which would be the case if the cell was incompressible. However, Poisson's ratio is rarely measured with biological materials, and the value of ν = 0.5 is assumed by analogy with synthetic compliant polymers and elastomers (4,23–26). Note that this value implies that the cell is incompressible, and although this may be a reasonable approximation of cell behavior, the laws of solid mechanics accept a value of 0.5 for Poisson's ratio as an upper bound to behavior if a material is isotropic and homogeneous, with real materials showing values slightly smaller than 0.5.

There have been a few published studies that have reported Poisson ratio values for single cells. Shin used an indentation method to compress individual osteoblast-like cells, and they reported a best fit to the data of ν = 0.37 ± 0.03 (27). In their study of cell deformation, Charras and Horton combined atomic force microscopy (AFM) with finite-element analysis to explore the influence of a Poisson ratio in the range 0.2–0.5 on their mechanical property measurements (28). Although no direct measurement of Poisson's ratio was made, Charras noted that the value of the Poisson ratio selected (assumed) made a significant difference to the apparent stiffness of the cell. Mahaffy et al. used AFM nanoindentation to determine the mechanical properties of NIH3T3 fibroblast cells using indentation models for thin compliant films bonded to a stiffer substrate. Their data are consistent with mechanical models using Poisson's ratio values in the range 0.4–0.5 (29). Although the studies of both Charras and Mahaffy report values for both Young's modulus and Poisson's ratio using a mechanical model of the indentation process, it is unlikely that these are truly independent measures, because in each case they are derived from a single mechanical measurement. Trickey et al. measured the mechanical properties of chondrocyte cells via micropipette aspiration and reported a Poisson’s ratio of ν = 0.38 ± 0.06 (30). Finally, Ma et al. studied the deformation of stem cell aggregates and deduced that a reasonable value is ν = 0.45 (31). Both Trickey and Ma used a finite-element model to predict deformation of the cells or cell aggregates in response to a force. In both these reports, Poisson's ratio was obtained through an optimization process in which the influence of changes in the value of Poisson's ratio and Young's modulus used in the model were compared to the overall observed deformation of a cell or cell aggregate. In all cases, the major criticism of the methodology is that it uses a single deformation mode to determine two independent elastic constants. From this published data, we conclude that there is no agreed value for Poisson's ratio of an individual cell, nor has there been any study of whether this elastic constant changes with induced changes in a cell's cytoskeleton.

Measuring the elastic properties of cells

There are a number of experimental techniques that can be used to measure the mechanical properties of cells, including aspiration of cells into capillaries (32), stretching of cells by optical tweezers (33), nanoindentation with AFM (23), and measuring acoustic wave velocity (34). Two independent mechanical measurements of a cell allow two independent components of the elasticity tensor to be determined. Here, the techniques we use are 1), measuring the acoustic wave velocity using scanning acoustic microscopy (SAM) images, and 2), nanoindentation using an AFM.

The combination of indentation and acoustic wave velocity measurements has been used previously to measure the elastic properties of homogeneous solid materials (35,36).

SAM is used to measure the mean longitudinal acoustic wave velocity, v_long, in a cell (34,37). This is directly related to C₁₁ by (16)

$$v_{long}=\sqrt{\left(\frac{C_{11}}{\rho }\right)},$$ (3)

where ρ is the mean density of the cell. The second experimental method analyzes the elastic response of a cell to spherical indentation carried out with an AFM, using Hertz’s solution for the indentation of an elastic half space by a sphere (23). An animal cell has an elastic stiffness that is many orders of magnitude smaller than that of most solid materials used in indentation experiments, and thus, the indentation can be modeled as if by a rigid sphere. In such a case, the displacement, δ, as a function of indentation load, P, is described by

$$\delta ={\left(\frac{9P^2}{16RE{\ast }^2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}.$$ (4a)

Here, R is the radius of the indenting sphere and E^∗ is the contact (or reduced) Young's modulus, defined by

$$E\ast =\frac{E}{1-{\nu }^2},$$ (4b)

where E and ν are the Young’s modulus and the Poisson’s ratio of the cell, respectively. In terms of the components of the stiffness tensor, Eq. 4b can be written as

$$E\ast =\frac{\left(C_{11}-C_{12}\right)\left(C_{11}+C_{12}\right)}{C_{11}}=\frac{\left(C_{11}^2-C_{12}^2\right)}{C_{11}}.$$ (5)

Measuring the acoustic wave velocity gives direct access to the C₁₁ component of the stiffness tensor (Eq. 3), which measures the resistance to volume change of a cell under pressure. AFM nanoindentation measures the contact Young's modulus of the cell, and as can be seen from Eqs. 2a, 2c, and 5, this is related to the difference between the C₁₁ and C₁₂ components of the stiffness tensor, which defines the resistance of the cell to shear deformation. Therefore, by combining measurements of the acoustic wave velocity in a cell with measurements of the contact modulus determined using the AFM, it is possible to measure a cell's resistance to shear deformation and volume change independently and to calculate from these data the isotropic elastic stiffness tensor of a cell and, hence, any of the cell's conventional elastic constants, without the need for prior assumptions.

Thus, by measuring two independent elastic constants, we provide better insight into the 3D mechanical response of a cell than can be achieved using a single elastic constant. This in turn allows the exploration of how different modifications of cell function separately affect cell resistance to change in volume and resistance to shear deformation. We illustrate this by investigating how the RhoA pathway influences cell mechanical behavior through the action of two downstream pathways that influence myosin-mediated cell contractility and formin (mDia)-mediated actin polymerization, both of which are important for coordinated cell motility (Fig. 1) (38,39).

Figure 1.

Schematic diagram, after Riveline et al. (21), showing the effect of Rho on cell mechanical properties through the activation of the ROCK-dependent and mDia1-dependent pathways. The ROCK-dependent pathway tensions the actin fibers through myosin-II-driven cell contractility either directly or indirectly via myosin light-chain phosphatase (MLCP). The mDia1-dependent pathway activates actin polymerization or, possibly, influences the organization of the actin network and microtubules. To see this figure in color, go online.

Materials and Methods

Cells, media, and reagents

NIH3T3 mouse fibroblasts (LGC-ATCC, Teddington, United Kingdom) were cultured at 37°C in Dulbecco’s modified Eagle’s medium (Sigma-Aldrich, Poole, United Kingdom) supplemented with 10% fetal bovine serum (Lonza, Cologne, Germany), 2 mM L-glutamine, and nonessential amino acids (all from Sigma-Aldrich). Nocodazole and ROCK-inhibitor Y-27632 were used at 1 μM and 100 μM, respectively (both from Sigma-Aldrich). For acoustic wave and AFM measurements NIH3T3 mouse fibroblasts were plated on glass-bottom dishes (MatTek Corporation, Ashland MA) that were coated for 1 h at room temperature with 10 μg/mL fibronectin diluted in phosphate-buffered saline (PBS) and rinsed twice with PBS.

Transfection

For transient transfections of NIH3T3 cell line, lipofectamine and Plus reagents (Life Technologies, Paisley, United Kingdom) were used according to the manufacturer’s instructions on cells seeded 24 h before transfection into six-well dishes. The plasmid encoding fluorescently labeled mDia1ΔN3 construct was from the A. Bershadsky lab (Weizmann Institute of Science, Rehovot, Israel). Glass-bottom dishes (MatTek, Ashland MA) were coated for 1 h at room temperature with 10 μg/mL fibronectin diluted in PBS and rinsed twice with PBS. At 33 h posttransfection, cells were plated in complete DMEM and allowed to spread in the glass-bottom dishes for 24 h before the experiment.

Immunofluorescence

After 24 h, cells were fixed with 4% (wt/vol) paraformaldehyde and permeabilized with 0.5% (vol/vol) Triton X-100 (both from Sigma-Aldrich) diluted in PBS. Samples were subsequently incubated for 1 h with an antibody directed against β-tubulin (clone DM1A, Sigma-Aldrich) directly conjugated with FITC. Staining for actin was performed at the same time using Texas-Red-X-conjugated phalloidin. Samples were observed on a Deltavision RT microscope (Applied Precision, Issaquah, WA) using a 100× NA 1.40 Uplan Apo objective (Zeiss, Cambridge, United Kingdom) and 86000v2 or 86006 filter sets (Chroma Technology, Bellows Falls, VT). Images were collected using a Coolsnap HQ CCD (Photometrics, Tucson, AZ) camera and converted and analyzed using Fiji software (40). Raw images were subjected to signal rescaling using linear transformation for display purposes in the figures.

Acoustic microscopy

Acoustic measurements were made using a KSI 2000 microscope (PVA TePla Analytical Systems; Herborn, Germany) with a custom data acquisition and control system. In brief, the system operates at 1 GHz, with quasimonochromatic tone bursts ≈20 ns in duration and a repetition rate of ≈500 kHz. The lens is initially focused at the substrate surface by monitoring the maximum output of the received signal with a gate setting optimized for in-focus signals. The lens was then raised 4 μm away from the substrate and a stack of images was taken at different z-positions commencing from this height toward the substrate surface with a step size of 0.1 μm over a range of 5 μm. It took ∼13 min to acquire a full stack of 50 images. The gray-scale value for every pixel (x, y position) was extracted from all the images at each z position to form a V(z) curve. Analyzing each V(z) curve provides a measure of the acoustic phase across the specimen, from which the mean acoustic wave velocity at each pixel location can be determined using a procedure described in detail elsewhere (37). The height of each cell was also measured using the interference rings visible in each acoustic image. The accuracy of this measurement was compared with AFM measurements of cell height and these were found to be consistent. Acoustic attenuation through a single cell is small and does not influence the acoustic wave velocity measurement. The acoustic wave velocity data was obtained by determining an average value from a linescan across the width of the cell nucleus region.

Atomic force microscopy

Cell elasticity was probed at the nucleus region of the spread-out cells with an AFM (Nanowizard, CellHesion 200; JPK Instruments, Berlin, Germany), installed on a fluorescence microscope (Axiovert 200M; Zeiss, Jena, Germany). Tipless AFM cantilevers (Arrow TL1, k = 0.03 N/m; NanoWorld, Neuchâtel, Switzerland) were modified by attaching 4.5-μm-diameter polystyrene beads at their ends. Modification of the cantilevers was done by Novascan (Ames, IA). Cantilevers were calibrated in the medium-filled chambers using the thermal noise method (41) before the experiments. Cells were indented at the nucleus region with an approach speed of 5 μm/s until a loading force of 2 nN was reached. Each cell was deformed in this way three times at the same position with ∼30 s intervals between indentations. The measured force-indentation depth curves (see Fig. S1 in the Supporting Material for an example) were fitted using built-in JPK software, implemented with the Hertz model for a spherical indenter, giving the Young’s modulus, E. The Poisson ratio of the cells was taken as 0.5 for the purpose of this calculation. Only the first 0.4 μm of indentation was used for determination of E; this is ∼10% of the height of the cells and was chosen to minimize any effect of substrate stiffness (42). In addition, only curves for which E was independent of indentation depth were used. For subsequent processing of data the influence of the assumed Poisson's ratio was eliminated by back calculating the contact modulus.

Statistical analysis

For both acoustic wave velocity measurements and contact modulus determination, a large number of experiments were repeated among the population of cells studied to determine the mean ± SE (indicated by the error bars). Statistical significance was determined using a two-tailed t-test with the level of significance set as probability p < 0.05.

Results and Discussion

Actin tensioning influences shear modulus but not bulk modulus

In non-muscle cells, the motor protein myosin IIA is identified as the myosin involved in actomyosin-regulated tension. Its activation state is regulated by phosphorylation (43) of its light chain, which is mediated by the Rho-associated protein kinase (ROCK) pathway (17,44). Downstream of RhoA, ROCK controls the myosin light chain kinase by regulating myosin light chain phosphorylation, which is important for actomyosin-mediated cell contractility (Fig. 1, MLCP). Another important effector of RhoA is mDia1, which when rendered active induces actin polymerization and increases microtubule stabilization. Thus, RhoA plays an important role in the regulation of cell mechanical properties, not only via the ROCK but also via the mDia1 pathway (Fig. 1) (38,45). To examine the contribution of ROCK to the stiffness of NIH3T3 cells, we blocked it by adding the protein kinase inhibitor Y-27632 (46). Through blocking of myosin II-mediated tensioning, treatment of cells with this drug led to the complete disassembly of actin stress fibers (Fig. 2, a and b).

Figure 2.

SAM and AFM experiments on populations of cells treated with Y-27632 and a control population. (a) Filamentous actin structure in a control NIH3T3 cell showing regular straight, tensioned filaments called stress fibers. (b) Highly disorganized actin filamentary structure was observed after treatment with Y-27632. (c) SAM image showing characteristic interference rings from a typical control NIH3T3 cell. (d) NIH32T3 cell after treatment with Y-27632, showing a larger number of interference rings and a more raised profile. Scale bars in a–d, 10 μm. (e) Mean (± SE) acoustic wave velocity (control, n = 27; Y-27632, n = 51) and contact elastic modulus (control, n = 92; Y-27632, n = 27). (f) Computed bulk modulus (K) and shear modulus (G). In all cases, the asterisk indicates a statistically significant difference (P < 0.05) between treated and control populations. To see this figure in color, go online.

Fig. 2, c and d, shows SAM images produced from NIH3T3 control cells and cells treated with Y-27632. The mean cell height (± SD) was 3.84 ± 0.68 μm for the control samples and 4.17 ± 0.77 μm after treatment with Y-27632, a difference that is statistically significant (p = 0.008). Parallel AFM measurements on a number of cells were carried out to determine cell height and these were consistent with the SAM data. Fig. 2, a and b, shows the corresponding fluorescence microscopy images of cells stained to reveal bundled filamentous actin; here, in the absence of tensioning, the actin is significantly more disorganized. The SAM data can be converted into acoustic wave (sound) velocity according to the procedure of Zhao et al. (37) and are shown in Fig. 2 e. For reference, the acoustic wave velocity in the PBS medium used for acoustic microscopy is ∼1490 ms⁻¹ (37), similar to the values of 1495 ± 7 ms⁻¹ and 1483 ms⁻¹ reported by Rapoport et al. (47) and Baddour et al. (48), respectively. Hence there is a significant difference between the acoustic wave velocity in the cell and that in the surrounding media.

The uncertainty in the exact height of a cell is probably the greatest cause of intrinsic error in measuring acoustic wave velocity. The error associated with uncertainties in cell height was calculated by Zhao (Fig. 8 in Zhao et al. (37)) and is ∼2 ms⁻¹ per 0.1 μm uncertainty in cell height for the cells in this study. Zhao reported a variation of about ±3 ms⁻¹ in the acoustic wave velocity measured across the nucleus region of a single cell, and we estimate that this is approximately what would be expected from uncertainties in local cell height. The standard deviation of the acoustic wave velocity measured in each cell population ranged from ∼12–23 ms⁻¹ (Table S1). Thus, the variation within each group of cells measured was always significantly greater than the intrinsic uncertainty in the method used to measure the acoustic wave velocity in an individual cell. To acquire the full data set, the cells are imaged over a period of ∼13 min. Because the elastic response is related to an acoustic excitation, the timescale of the mechanical response is unrelated to the time taken for image acquisition.

Our acoustic data give results similar to the acoustic wave velocities obtained by Kundu et al. for XTH2 cells using 1 GHz acoustic waves with a different analysis methodology (1500–1700 ms⁻¹) (34), and by Taggart et al. for a range of human-derived cells (1522–1537 ms⁻¹) using 40 MHz acoustic waves (49). Our improved analytical method uses phase information and has been shown to reduce the scatter and uncertainty in the acoustic wave velocity compared with other published methods, hence explaining the lower levels of experimental scatter when our data are compared with those of Kundu or Taggart (37). We note that the addition of Y-27632 does not significantly change the acoustic wave velocity (p = 0.3).

The AFM measurements (Fig. 2 e) show a significant decrease in the contact modulus after the addition of Y-27632 (p < 10⁻⁶). This apparent discrepancy with the SAM data may appear surprising, but we will show that this is consistent with the tensorial representation of the elastic properties of cells. Note that the statistical scatter of the two measurements are quite different, with the acoustic wave velocity measurement showing a much lower variation as a fraction of the measurement value than the corresponding AFM data. We interpret this as demonstrating a greater intrinsic variability in the AFM measurement of cell stiffness. This is almost certainly caused by error in identifying the contact between the AFM tip and the cell surface, which will lead to uncertainties in the precise value of δ in Eq. 3a (50). When the tip is immersed, there are further hydrodynamic effects that may additionally impede surface detection (51). Further uncertainties may occur when determining the AFM cantilever spring constant. We have calculated the cantilever spring constant using the thermal fluctuation method (41), as used by many research groups, however this may lead to errors in excess of 30% of the measured value (52). Note that these errors will be between groups, because a single AFM cantilever was used to measure each individual cell population studied.

The full dataset used in this study and in the studies subsequently reported are presented in Table S1. If we compare our AFM cell stiffness data with that in the literature (a convenient review is presented by Kuznetsova et al. (53)), it is clear that our data (control cell population Young's modulus of E = 1520 ± 646 Pa (mean ± SD)) are similar in both magnitude and reported experimental scatter to those reported in other AFM studies of the stiffness of fibroblast cells.

If the density of the cell is known, our data can be used with Eqs. 2–5 to determine the two independent terms of the isotropic stiffness tensor (see Table S2). There have been relatively few studies of the density of individual cells under physiological conditions, chiefly because of the difficulty of making such measurements. Anderson et al. used high-molecular-weight solute to vary the density of cell-suspending media with minimal effect on osmotic pressure and used a range of such fluids to study the centrifuge-assisted migration of cells (54). This method was used to study how the density of CHO cells varied through the cell cycle, to obtain an estimated density (± SD) of 1052 ± 3 kg m⁻³ with any variation in density during the cycle being significantly lower than the standard deviation. Anderson et al. used their calculations to comment on density measurements by other workers, who had used similar methods but had not corrected for osmotic effects. They concluded that because the density measurements obtained for rabbit thymocytes, LKID, and HeLa cells were all within 6 kg m⁻³ of the CHO data, there was considerable uniformity in the density of mammalian cells. Grover et al. have developed a method to measure the mass of individual cells using a micromachined resonator and coupled this with optical measures of cell volume to calculate cell density (55). Unfortunately, their published data only report the density of erythrocytes and lymphocytes.

The density of a cell measured in suspension may be different from that of a cell adhered to a substrate, but in the absence of any data to confirm this we have used the values determined from isolated cells in suspension. Thus, on the basis of Anderson's analysis, we assume a constant cell density value of 1050 kg m⁻³ throughout this study. We use this to compute the engineering elastic constants of the shear and bulk moduli, G and K, respectively (Fig. 2 f). The error values of the elastic constants were calculated using conventional error propagation laws from the standard error reported for the data obtained using AFM and acoustic microscopy (Table S2). We believe that any systematic error from differences in the values of the cell density is unlikely to be much larger than the population error.

Here, we note that we are constructing a stiffness tensor using data measured with deformation time constants that differ by several orders of magnitude. It is well known that viscoelastic materials show a time dependent deformation behavior that can result in quite different values of the apparent Young's or shear modulus depending on the timescale of the deformation. However, the bulk modulus shows a much smaller influence of deformation rate (56). This is probably because the mechanism for viscoelastic behavior is related to polymer deformation through chain sliding. If the deformation is primarily a volume change, there will be no mechanism for polymer chain sliding and the elastic properties will be independent of loading rate. We postulate that because the acoustic wave velocity is related to the C₁₁ modulus (Eq. 3), it too will show little variation with loading rate, and thus, we can combine the AFM data with the acoustic microscope data, despite the fact that the acoustic system operates at a rate of 1 GHz and the AFM data are acquired at a rate in the effective range 1–10 Hz.

The change in the bulk and shear modulus illustrates how the action of Y-27632 influences the cell’s resistance to volume change and shear. From these figures, we can see that the resistance to volume change is unaffected within statistical limits, but the resistance to shear has been reduced dramatically. This behavior is interpreted as an effect of the deactivation of the myosin motors, which eliminates the tensioning of the actin filaments but neither influences the quantities of the cytoskeleton structural polymers (actin and microtubules) nor promotes any structural change (e.g., cross-linking) and thus does not change the elastic compressibility. However, removing the tensioning of the actin filaments has allowed the deformation of the cell to occur easily under shear loading with no change in volume. This is in agreement with data from in vitro experiments on f-actin and myosin II gels by Mizuno et al., where shear stiffness was only present when the actin filaments were tensioned by the myosin (57). Such behavior is fully consistent with a tensegrity model for the cytoskeleton, where structural integrity is maintained by tensioning of structural fibers (19). In the absence of such tensioning, there will be significantly reduced resistance to shear (shape change), whereas there will be no difference to any resistance to change in volume.

Actin polymerization influences both shear and bulk modulus

To examine the effect of increased actin polymerization on NIH3T3 cells, we expressed a constitutively active form of the formin mDia1 (mDia1ΔN3) in NIH3T3 cells, which has been shown to dramatically promote actin polymerization in cells (45,58). Cells expressing mDia1ΔN3 have a characteristic bipolar shape, with the dense actin filaments following the long axis of the cell (Fig. 3 a). The characteristic shape of the mDia-expressing cells can be readily identified by the SAM image (Fig. 3 b); in this case, the cells have a mean height of 4.12 ± 0.81 μm, which is slightly higher than the mean height of the control population (p = 0.033). In Fig. 3 c, the transfected cells show a small but statistically significant increase in acoustic wave velocity (<20 ms⁻¹; p = 4.9 × 10⁻⁴). This is consistent with previous in vitro studies of actin gels (59), where small increases in acoustic wave velocity were observed with increasing polymerization of dilute actin gels. The transfected cells also display a significant (and relatively larger) increase (p = 2.0 × 10⁻⁴) in the contact modulus (Fig. 3 c). These data have been used to compute the bulk and shear moduli of the control and transfected cell populations (Fig. 3 d). From this, it is clear that the mDia1ΔN3-induced actin polymerization has resulted in a significant increase in both the bulk and shear moduli. We interpret this result as showing that by increasing the filamentous fraction of actin, the cell becomes more rigid and less compressible. This is supported by the predictions obtained using classical rule-of-mixtures estimates of the elastic properties of composites (60), where an increase in the fraction of the stiffer component in a composite structure (in this case actin filaments) leads to an increase in both the shear modulus and Young's modulus.

Figure 3.

SAM and AFM experiments on cells transfected with mDia1ΔN3 and a control population of cells. (a) NIH3T3 cell transfected with mDia1ΔN3 packed with a dense network of actin stress fibers; scale bar, 10 μm. (b) SAM image showing interference rings from transfected NIH3T3 cells; scale bar, 20 μm. (c) Mean (± SE) acoustic wave velocity (control, n = 38; mDia1ΔN3, n = 45) and contact elastic modulus (control, n = 30; mDia1ΔN3, n = 55). (d) Computed mean ± SE bulk modulus (K) and shear modulus (G). In all cases, the asterisk indicates a statistically significant difference (P < 0.05) between treated and control populations. To see this figure in color, go online.

Influence of microtubule disruption countered by actin polymerization

Nocodazole is commonly used to disrupt microtubules. Thus, one might expect that disrupting microtubules in cells would reduce cell stiffness and shear resistance (19). Fig. 4, a and b, shows that the network of microtubules revealed through staining a control cell population has disappeared after treatment with nocodazole. Fig. 4, c and d, shows that the actin stress fibers are slightly increased by the action of nocodazole, but in this case, the mean height of the cell in the SAM images (Fig. 4 e) is 2.72 ± 0.49 μm and significantly smaller than the control cell population (p < 10⁻⁶). Despite the disruption of microtubules, there is a small but statistically significant increase in acoustic wave velocity (p = 5.9 × 10⁻³) and also a significant increase in contact elastic modulus (p = 1.0 × 10⁻⁵) (Fig. 4 f). From these data, it can be seen that both the bulk and shear modulus values also show a significant increase (Fig. 4 g). These results are similar to those reported by Wu (11), who also measured an increase in cell stiffness in the presence of nocodazole. It has been shown that the disruption of microtubules leads to the activation of RhoA (61), thereby activating downstream ROCK, which leads to myosin activation, and endogenous mDia, which leads to increased actin polymerization (Fig. 1). It is thus these mechanisms that seem to compensate for the presumed reduction in cell stiffness that should result from microtubule disruption. We note that in the case of the nocodazole-treated cells, the AFM indentation, 0.4 μm, is >10% of the cell height at the nucleus, and hence, the increase in stiffness recorded by the AFM may be partly caused by the influence of the substrate sensed when indenting thin sections (42).

Figure 4.

SAM and AFM experiments on NIH3T3 cells treated with nocodazole and a control population. (a) NIH3T3 cell in control conditions stained to reveal microtubules. (b) After treatment with nocodazole, the microtubule structure is no longer revealed by the stain, indicating severe disruption. (c) NIH3T3 cell in the control condition stained to reveal actin. (d) After treatment with nocodazole, actin filaments are clearly present in an amount similar to that observed in the control population and are well aligned. Scale bars in a–d, 10 μm. (e) SAM image of the cells after treatment with nocodazole. The presence of several interference rings indicates that the cells have maintained a 3D shape; scale bar, 20 μm. (f) Mean (± SE) acoustic wave velocity (control, n = 34; nocodazole, n = 47) and contact elastic modulus (control, n = 39; nocodazole, n = 40). (g) Computed mean (± SE) bulk modulus (K) and shear modulus (G). In all cases, asterisks indicate a statistically significant difference (P < 0.05) between treated and control populations. To see this figure in color, go online.

Poisson’s ratio of a cell is very close to 0.5

Finally we consider the Poisson’s ratio of the cells. Most continuum models of cell mechanical behavior take the Poisson’s ratio of the cell to be ν = 0.5, which assumes the cell to be incompressible. Indeed, most measurements of the Young’s modulus of a cell using Hertzian indentation models have taken this value of Poisson’s ratio (4,23–26). A small number of studies have assumed a different value, with Charras and Mason (28) and Kelly et al. (62) using ν = 0.3 and ν = 0.37, respectively.

Our results show that the cell has a finite bulk modulus (see Figs. 2 f, 3 d, and 4 g), albeit orders of magnitude greater than the shear modulus. All our results indicate that K >> G for all control cells and treated cells investigated, hence from Eq. 2, we expect Poisson’s ratio to be very close to 0.5. Thus, to explore the variation in Poisson’s ratio, ν, it is more convenient to plot the deviation of its calculated value from 0.5. Fig. 5 shows the value of 0.5 − ν for all the cell populations studied. The error value reported for our Poisson's ratios was determined using the standard laws for error propagation (Eqs. 2–5). This apparently low value can be better understood with reference to Eq. 2b, where it is clear that any deviation from the incompressible value of ν = 0.5 is a measure of the difference between C₁₁ and C₁₂. The difference between these elastic constants is independent of the measured acoustic wave velocity and is instead determined by our AFM data for E^∗, as seen clearly in Eq. 4. Thus, the error in our estimation of Poisson's ratio is related to the error in measuring E^∗ divided by the value of C₁₁.

Figure 5.

Deviation of the value of the Poisson ratio from the incompressible limit (ν = 0.5) for cell populations studied after a number of treatments.

Poisson's ratio was calculated for the control and treated populations for all three cell treatments studied. In all cases, the measured Poisson’s ratio is very close to 0.5, with (0.5 − ν) < 1.5 × 10⁻⁷. The addition of Y-27632 leads to the value closest in magnitude to 0.5, consistent with the very low resistance to shear induced by the absence of myosin activity. Indeed, from Eqs. 2b and 2c, it is clear that if ν = 0.5, then G = 0, which is a characteristic feature of a liquid. There is to our knowledge only one published study, by Maniotis et al (63), on the influence of cell treatment on Poisson's ratio. The authors state that they measure an apparent Poisson's ratio by anchoring a cell through puncturing the membrane before capillary deformation and subsequently measuring the displacement within an area of ∼10 μm². They find a low value of the apparent Poisson's ratio (∼0.25) in a control population that increases dramatically when cytoskeleton-modifying drugs are added, in contrast to the very small change in value reported here. This discrepancy may in part be explained by the very large displacements used in Maniotis's study, in which case some of the deformation may be outside the elastic regime and represent permanent deformation of the cell. Indeed, the authors comment that some of their observed change in Poisson's ratio can be explained by fluid loss from the cell. Our Poisson's ratio data indicate that cell volume is conserved during deformation, and this behavior is consistent with a high fluid content within the cell.

Conclusions

To model the behavior of a cell as an elastic body requires a minimum of two elastic constants, even if the cell is represented as a homogeneous, isotropic continuum. These two properties are needed because the small-scale elastic deformation of a body can be decomposed into a change in volume and a change in shape. By using two independent methods, SAM and AFM, to characterize the mechanical behavior of a population of cells, the first method sensitive to changes in volume and the second to changes in shape, we have been able to determine two independent components of the stiffness tensor of a cell type and hence calculate two independent elastic constants without any prior assumptions concerning their mechanical behavior. Using this approach, we were able to determine accurately the following information about the resistance of cells to relatively small-scale deformations with time constants <1 s.

We have demonstrated that Poisson's ratio of NIH3T3 cells is very close to, but slightly less than, 0.5, deviating from this value by <1.5 × 10⁻⁷ for all conditions studied. This shows that a cell is very close to incompressible in behavior, which is consistent with a high free-flowing fluid level in the cytoplasm.

By monitoring the changes in the two elastic properties after modifying the cytoskeleton of NIH3T3 cells, we identify how changes to the composition and organization of the cytoskeleton contribute to the resistance of a cell to deformation by separating the effect on volume change from that on shape change. The bulk modulus of the cell depends on the quantities of the structural components of the cytoskeleton and is sensitive to the total actin filament content. However, to resist a change in shape (shear deformation), the actin filaments must be under tension, and in the absence of tension, the cell shows a dramatic reduction in shear modulus but not in bulk modulus. A similar behavior is seen with the Poisson's ratio measured for the cell. In all cases, this value is slightly smaller than 0.5, but it approaches 0.5 when the actin fibers are not tensioned. A Poisson's ratio of 0.5 is accompanied by a shear modulus of zero and is thus incompatible with any resistance to shape change. These results are consistent with the tensegrity model for the influence of the cytoskeleton on cell stiffness (19). However, by providing evidence for a structure consisting of a rigid tensioned framework within a liquid cytoplasm, our data also suggest that a more sophisticated model for cell behavior, which would be needed for large-deformation or long-time studies, would need to account for the redistribution of fluid within a deforming cell, and this would likely be a biphasic or poroelastic constitutive law.

This study shows that by combining two independent probes of the mechanical properties of cells we can provide a more complete picture of how changes in the cell cytoskeleton influence its mechanical behavior. The approach we use assumes a simple elastic model of cell behavior, but by identifying two independent components of the stiffness tensor, it allows for the 3D nature of mechanical interactions of cells with the environment. We believe that this more complete description of cell mechanical behavior will lead to a better understanding of the pathology of disease and cancer at the cellular level.