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

Abstract

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

Introduction

Mechanical properties of cells are important factors defining cell functionality, motility, tissue formation (1,2), stem cell differentiation (3), etc. Changes in cell elasticity as a marker for cell abnormalities, and a correlation with various human diseases, has been recently discovered. It has been implicated in the pathogenesis of many progressive diseases, including vascular diseases, kidney disease, cancer, malaria, cataracts, Alzheimer’s, diabetic complications, cardiomyopathies, etc. (4–6). In some cases, it is believed that the loss of tissue elasticity arises from the changes in the extracellular matrix (7), not in the cells per se. However, it has also been shown that the cells themselves can also change their elasticity quite considerably due to cancer, malaria, arthritis, and even aging (8–10). Furthermore, the stiffening of red blood cells infected with malaria (11,12) was found to be responsible for fatal incidents of this disease. Low rigidity of cancer cells was recently suggested as an indicator for cancer diagnosis (13,14). Therefore, in addition to the fundamental interest, there is a practical need to measure cell mechanics quantitatively.

At the same time, a number of experimental results show complexity and sometimes ambiguity in the obtained results. For example, in contrast with the low rigidity of cancer cells reported in the majority of works, there are results demonstrating no change (15) or even increase of rigidity (16) with malignant development. Another example is related to a viscoelastic response of cells. Cells typically demonstrate higher rigidity (storage and instantaneous modulus) with the load rate increase (17). However, such behavior was not observed in the other work (18). Thus, it is important to test validity of the models used to derive the quantitative mechanical properties of cells.

To have the measurements interpreted in a quantitative way, one needs to characterize mechanical properties in an instrument/method/model-independent way. This is typically done with the help of elastic moduli (19), quantitative parameters assigned to material, not the way it is measured. It should be noted that the cell is a rather complex object. Although it is known that the majority of complex structures can be characterized with the elastic moduli when the contact stresses and strains are sufficiently small, it is questionable whether cells can be described in terms of the elastic modulus at all in a self-consistent (quantitative) way. This work is an attempt to answer this question.

There are three primary static moduli of elasticity that can be used to describe the cell: Young’s (tensile), shear, and bulk. Assuming a cell is a homogeneous and isotropic material (at least for relatively small indentations), the cell can be characterized by just two parameters—for example, by the elastic modulus and the Poisson ratio (19). It should be noted that the term “elastic modulus” exclusively refers to the Young’s modulus in this work. It is done for consistency with our previous works and to address the existing concern that the Young’s modulus might require redefinition at the nanoscale. Because the Poisson ratio of soft materials typically ranges within 0.3–0.5 (20,21), the maximum error in the definition of the elastic modulus due to the unknown Poisson ratio is expected to be <10% (22). Therefore, it makes sense to characterize mechanics of cells with just one parameter, the elastic modulus.

The atomic force microscopy (AFM) technique has become popular in the study of cells (15,23,24). In particular, it is possible to use the AFM probe to indent a cell, to study cell mechanics by recording the cantilever deflection while deforming the cell (25,26). The indentation can be measured while the cell is being immersed in physiological media. The lateral position of the AFM probe and the load force can be controlled with rather high precision (27,28). This positioning can be done on either individual cells or cell layers in a culture dish (in vitro), or even on pieces of tissue (ex vivo) (29).

To derive the elastic modulus from the indentation experiments, simple mechanical models are typically used—the Hertz, for a spherical indenter; and Sneddon, for a conical indenter. We recently suggested an extension of these models to the case of soft samples in which the indenter starts to interact with the cell surface before contacting it physically (30,31). As was demonstrated, the majority of cells have a surface covered with various membrane protrusions and corrugations (microvilli, microridges, filopodia) and a glycocalyx, which we collectively called a pericellular “brush.” The brush, at least partially, can be observed by means of electron and confocal optical microscopy, see, e.g., Iyer et al. (15). Although the brush is there, we consider the models with no brush in this work because 1), those models are the ones in most common use, and 2), the brush contribution to the cell mechanics might be insignificant. All four models (Hertz/Sneddon, with/without brush) assume constancy of the elastic modulus. This assumption can be verified, for example, by calculating the elastic modulus for different depths of indentation. Observation of such dependence would mean inconsistency of the model.

In this work, we verify the consistency of the above elastic models by testing the dependence of the elastic modulus on the indentation depth. To amplify, the independence of the elastic modulus of the indentation depth is the necessary condition of applicability of any model in which the material is considered elastic and homogeneous/isotropic. Substantially different AFM indenters (conical sharp and spherical dull ones) were used to collect the indentation data. Analyzing a statistically sound amount of data, we found that independence of the elastic modulus of the indentation depth could be observed only for the indentation done with the dull spherical probes when processing the indentation data with the brush model. For all other three models, when data was processed with Hertz/Sneddon models (no brush taken into account), and with the brush model when data was collected with sharp conical probes, there was significant depth dependence observed. One can conclude that the only self-consistent approach to derive the elastic cell modulus is to use the brush model to process the indentation data collected with a dull AFM probe (of well-defined geometry). Upon studying the cell mechanics in this way, it seems possible to describe the elastic properties of the cell body by an effective Young’s modulus (the elastic modulus) in a self-consistent manner.

Materials and Methods

Cells

Primary cultures of human epithelial normal and cancer cells were collected from tissue of the cervix of healthy and cancer patients, respectively. All human tissue was obtained from the Cooperative Human Tissue Network (CHTN, National Cancer Institute, National Institutes of Health, Bethesda, MD). Informed consent was obtained from patients according to their published guidelines (http://chtn.nci.nih.gov/phspolicies.html). The cells were prepared by a two-stage enzymatic digestion of cervical tissue, as described in Gaikwad et al. (32) and Woodworth et al. (33,34). Briefly, each tissue was digested for 16 h at 4°C in dispase and the layer of epithelial cells was removed from the underlying connective tissue by scraping. The sheet of epithelial cells was cut into 1 mm² pieces and digested in 0.25% trypsin at 37°C for 10 min. Trypsin was neutralized by adding fetal bovine serum, and cells were collected by low-speed centrifugation. Cultures consisting of ≥95% epithelial cells were maintained in keratinocyte serum-free medium (Invitrogen, Carlsbad, CA), which prevents outgrowth of fibroblasts and other stromal cells (no evidence of contamination by fibroblasts or other stromal cells was observed).

Cells were placed in 60-mm culture dishes and feed three times per week with keratinocyte serum free medium (Invitrogen). Normal cell strain typically consisted of cells that had been maintained for <3 passages in vitro (40–60 population doublings), when they were actively growing, and carcinoma cell lines were used at population doublings 60–120. The higher number of divisions of cancer cells was used to avoid possible confusion between cancer and normal cells (possibly normal cells present in the cancer culture dish would die out before that number of population doublings).

All cells were plated in 60-mm tissue-culture dishes, and dishes were used for experiments when cells were <50% confluent. Right before imaging, the cells were twice washed with Hank’s Balanced Salt Solution medium (HBSS; Life Technologies, Carlsbad, CA) and imaged in this medium. A quantity of 30–40 cancer and normal cells were used for the indenting experiments with either sharp or dull probes.

Atomic force microscopy

Indenting Speed

A Bioscope Catalyst AFM (Bruker/Veeco, Billerica, MA), placed on a model No. TE2000U confocal Eclipse microscope (Nikon, Melville, NY), and a Dimension 3100 AFM (Bruker/Veeco) with NPoint close-loop scanner (200 × 200 × 30 μm, XYZ) were used in this study. Dimension AFM was equipped with a built-in video microscope that helps with positioning the AFM probe about the cells of interest (allows observation of areas from 150 × 110 to 675 × 510 μm² with 1.5-μm resolution).

Standard cantilever holders for operation in liquids were employed. The force-volume mode was used in this study. In this mode of operation, an AFM probe moves up and down recording the force-indentation curve at each pixel of the surface. After recording each force curve, the AFM probe moves up, and then is displaced in the lateral direction to the next pixel of the surface to continue force recording. The force-volume images of cells were collected with the resolution typically of 16 × 16 pixels within 50 × 50 μm² area. The force-volume allows simultaneous recording of the cell topography and force-indentation curves at each pixel of the surface. The force-indentation curves are analyzed only from a relatively flat area above the cell nucleus (the incline is smaller than 10–15%).

Standard V-shaped arrow 200-μm AFM cantilevers (Bruker, Santa Barbara, CA) with integrated silicon nitride pyramidal probes (sharp probes) were used. Spherical colloidal probes were prepared as described, in detail, in Berdyyeva et al. (25) and Volkov et al. (35). Briefly, these tipless Bruker cantilevers were used to glue 5-μm diameter silica spheres (Bangs Laboratories, Fishers, IN) to the cantilevers attached to either the AFM built-in micromanipulator (Dimension 3100 microscope; Bruker/Veeco) or a Micromanipulator Station 6000 (Micromanipulator, Carson City, NV). The radius of the probe was measured by imaging the inverse grid (TGT1; NT-MDT, Moscow, Russia). The cone half-angle of the sharp probe (22.5°) was measured with either the help of electron microscopy or the same inversed grid. The cantilever spring constant (0.04–0.3 N/m) was measured using the thermal tuning method (the algorithm from the built-in AFM software) before gluing the spherical probe.

Measurements of the static elastic modulus imply the use of small indentation speed in the AFM experiments. This is not easy to do when dealing with soft materials in general because of the creep, a slow increase of indentation depth under constant load. In the case of biological cells, this issue is even more complicated because slow indentation can induce a nontrivial biological response from a cell. For example, cells may start to restructure their cytoskeleton, or develop nontrivial adhesion to the AFM probe (32,37,38), or simply crawl away.

As a compromise, the force-indentation curves are typically recorded with a ramp frequency of 1–2 Hz with the vertical ramp size of 3–6 μm. This reasonably minimizes the viscoelastic effects of the indentation, although a difference between approaching and retracting force-indentation curves is still seen even at these speeds. Therefore, it is important to consider similar speeds when comparing the results of different works. Here we used 10 μm/s ramping speed (ramp frequency of 1.2 Hz with the vertical ramp size of 4 μm).

Models used to derive the elastic modulus of cells

Four models are used in this work: the Sneddon model (the case of a cone indenter) (39), the Hertz model (the case of a spherical indenter) (40), and the brush models for either conical (derived in this work) or spherical indenters (31,41). All these models were derived under the assumption that the material of study can be characterized with just one elastic modulus, which is indentation-independent. It is definitely a strong assumption for such a complex object as a cell. Nevertheless, if this assumption is not valid, strictly speaking, one cannot use the above models. Thus, it is important to test the independence of the elastic modulus of the indentation depth to verify self-consistency of these models.

The Hertz model and its various modifications (25,42–46) have been widely used to determine the elastic modulus of cells. In these models, the cell is assumed to be a homogenous material, and the cell border is a well-defined interface. Although homogeneity of the cell material may be considered as a reasonable approximation for small deformations, the cell surface is far from being a well-defined flat interface because it is covered with the cellular brush. The Hertz and Sneddon models are still broadly used to calculate the elastic modulus of cells. Therefore, we will study self-consistency of those models as well.

The model that takes into account the presence of pericellular brush (15,41) allows deriving the elastic modulus of the cell body as well as the parameters of the pericellular brush. The brush model for a spherical indenter is described in detail in Sokolov et al. (47). The case of a conical indenter is developed in this work. It should be noted that an attempt to derive a brush model for a conical indenter is found in Wang et al. (48). However, the formula for the elastic modulus was misprinted, and the formula for the brush force was used for the spherical, not conical indenter in that work. Here we will briefly overview the brush model for a spherical indenter, derive formulas for a conical indenter, and extend the brush model to a broader range of the indentation depths.

It should be noted that all models mentioned above have been developed for an indenter over either plane or (hemi)sphere. Thus, we processed only the force curves from the top area of the cell, which can be approximated as a hemisphere. Following the previous works (10,15), we take the force curves in the surface points around the top when the incline of the surface is <10–15° (the spherical approximation of the cell is reasonable for this small area). To identify such curves, the AFM image of cell heights was used (the height image was collected as a part of the force-volume data set). The effective radius of the cell was derived from these images after taking into account the cell deformation (value i in Eq. 3). Below, we briefly describe the models used in this work.

The Hertz model

The Hertz model is widely used in the literature for a spherical shape indenter. In this model, the cell is treated as a homogenous smooth (well-defined boundary) semisphere of radius R_cell. To derive the elastic modulus of the cell, the experimental force-indentation curves are fitted with

$$F\left(i_c\right)=\frac{16}{9}E\sqrt{\frac{R_{\text{probe}}\cdot R_{\text{cell}}}{R_{\text{probe}}+R_{\text{cell}}}}{i}_c^{3/2},$$ (1)

where F is the load force, E is the elastic modulus of the cell, i_c is the indentation depth, and R_probe is the radius of the apex of the AFM probe. Here the Poisson ratio of the cells is set to 0.5.

The Sneddon model

A more general case of geometry of the indenter is described by the Sneddon model (39). The most popular case of the conical indenter of semi-angle α is traditionally used. Here the cell is also treated as a homogenous smooth (well-defined boundary) medium. To derive the elastic modulus of the cell, the experimental force-indentation curves are fitted with

$$F=\frac{8}{3\pi }E\tan \alpha \cdot i_c^2,$$ (2)

where F is the load force, E is the elastic modulus, and i_c is the indentation depth. Here the Poisson ratio of the cells is set to 0.5.

Brush model: spherical indenter

If one considers a cell as a spherical object covered with a brush layer, the AFM probe squeezes both the cell body and brush at the same time (Fig. 1). To separate the elastic deformation of the cell body from the deformation of the brush layer, the following model was suggested in Sokolov et al. (41). A geometrical consideration (see the notations defined in Fig. 1) gives the equation

$$h=Z-Z_0+i+d.$$ (3)

The relative piezo position of the cantilever Z and the cantilever deflection d are directly measured with AFM when collecting the force-load curves (so-called raw data). The other two parameters, deformation of the sample i, and nondeformed position of the sample Z₀, must be found. The elastic modulus is included in the deformation of the cell body i.

Figure 1.

A schematic of AFM probe-cell surface interaction. Brush layer is shown. The value Z is the relative piezo position of the cantilever, d is the cantilever deflection, Z₀ is the nondeformed position of the sample surface, i is the deformation of the sample, and Z = 0 is for the maximum deflection assigned by the AFM user. The value h is the separation between sample and the AFM probe.

A substantial simplification comes from the assumption that the brush is softer than the cell body. The validity of this assumption can be confirmed experimentally, or through self-consistency of this model; see the explanations below. Using this assumption, one can unambiguously fit the experimental data with the parameters of Eq. 3, the elastic modulus E, and nondeformed position of the sample Z₀, by considering the limit of squeezed brush. Technically, it is done by saying that h → 0 somewhere before reaching the maximum load. Obviously, this assumption also depends on the value of the maximum load. We treat it as plausible because 1), it is the parameter one can control with the AFM setup, and 2), it can be directly checked retrospectively (finding h after calculating the elastic modulus). This self-consistency of the squeezed brush will be checked retrospectively for all calculations. This will be exampled and discussed below and in the Results and Discussion.

Assuming that one can squeeze the brush to the list h → 0 and using the Hertz relation between the indentation of the elastic part of cell, i, and force F (Eq. 1), one can arrive at

$$Z_0-Z={\left[\frac{9}{16}\frac{k}{E\sqrt{R^{\ast }}}\right]}^{2/3}{\left(\frac{F}{k}\right)}^{2/3}+\frac{F}{k}.$$ (4)

Here E is the elastic modulus of the cell body, the Poisson ratio ν of the cells is chosen to be equal to 0.5, k is the spring constant of the AFM cantilever, and R^∗ = R_probe ⋅ R_cell/(R_probe + R_cell) R_probe. The value R_probe can be found before starting measurements as described above. R_cell can be restored from the topographical image of the cell obtained in the force-volume mode corrected by the cell body deformation i.

After finding the elastic modulus, one can separate the contribution of the brush by finding the force dependence due to brush (F versus h) from the recorded force curves. The sought force-dependence d(h) can be found from Eq. 4 by using the found elastic modulus E as follows:

$$h\left(d\right)=Z-{\left[\frac{9}{16}\frac{k}{E\sqrt{R^{\ast }}}\right]}^{2/3}\left(d_{\max }^{2/3}-d^{2/3}\right)-\left(d_{\max }-d\right).$$ (5)

The force F(h) caused by the existence of the brush can now be reconstructed using the following formula: F(h) = kd. It is instructive to use the model of entropic brush, which gives a way to introduce the following brush parameters quantitatively: effective grafting surface density of the brush constituents (grafting density) N and the brush length L (41,49), as

$$F\left(h\right)\approx 50k_{\text{B}}T{R}^{\ast }{N}^{3/2}\exp \left(-2\pi \frac{h}{L}\right)L,$$ (6)

where k_B is the Boltzmann constant, and T is the temperature. This formula is valid for 0.1 < h/L < 0.8 (50).

Brush model: modification for conical indenter

Here we derive the brush model for the case of the conical indenter. It is rather close to the previously described case of the spherical indenter. The main difference is in Eqs. 4 and 5, in the part that describes the deformation of the cell body with the indenter. To describe the indentation of the cell body with the squeezed brush, the Sneddon model for a conical indenter of semi-angle α should be used instead of the Hertz model. Equation 4 is then written as

$$Z_0-Z={\left[\frac{3\pi F}{8E\tan \alpha }\right]}^{1/2}+\frac{F}{k}.$$ (8)

Similarly, Eq. 5 reads

$$h\left(d\right)=Z-{\left[\frac{3\pi k}{8E\tan \alpha }\right]}^{1/2}\left(d_{\max }^{1/2}-d^{1/2}\right)-\left(d_{\max }-d\right).$$ (9)

The equation for the brush force can be derived by following the recipe described, e.g., in Butt et al. (51) and Sokolov (52). Specifically, the force can be found as an integral used in the derivation of the Derjaguin approximation:

$$F\left(h\right)\approx 100\pi k_{\text{b}}T{N}^{3/2}\underset{h}{\overset{\infty }{\int }}\exp \left(-\frac{2\pi D}{L}\right)\left(D-h\right){\tan }^2\alpha dD.$$ (10)

Taking this integral, one obtains

$$F\left(h\right)\approx \frac{25}{\pi }{k}_{\text{B}}T\cdot N^{3/2}\cdot {\tan }^2\alpha \cdot \exp \left(-2\pi \frac{h}{L}\right)L^2.$$ (11)

This formula is also valid for 0.1 < h/L < 0.8.The steps of extracting the elastic modulus and brush parameters in this model are identical to the above case of the spherical indenter.

A note about verification of the degree of the brush compression assumed in the brush models

Equations 6 and 11 allow testing of the degree of the brush compression, which is assumed during derivation of the elastic modulus in the brush models. This is done retrospectively after finding the brush parameters as described above. The question, however, is what degree of compression should be considered as sufficient. One can note that Eqs. 6 and 11 lose their validity when the AFM probe-cell surface distance h is <10% of the brush length L (some sources give 20%). When the brush is squeezed to a higher degree, the brush behavior turns into the behavior of an elastic layer. Furthermore, one can easily see that the effective stiffness of the brush is increasing with the brush compression. At one point, the stiffness of the substrate will be equal to the stiffness of the squeezed molecular brush, and therefore, the elastic responses will become similar. Thus, the error due to the deviation h from zero can be assigned to the uncertainty in the indentation depth. For example, if we consider 90% deformation of the brush as a good approximation of completely squeezed brush, this results in 15% maximum error in the definition of the elastic modulus. This is quite acceptable for this level of accuracy of the AFM quantitative analysis. In principle, this can further be improved if needed.

Results and Discussion

Let us demonstrate the analysis of the collected force-indentation curves through either Hertz/Sneddon models or the brush models. An example of processing raw AFM data, deflection of the AFM cantilever versus vertical position of the AFM scanner (d versus Z) is shown in Fig. 2. The approaching part of the force-indentation curves was analyzed. The data shown were collected with a spherical AFM indenter. Fig. 2 a shows the regions of the curve fitted with the equations of the Hertz and brush models. The fitting region in the Hertz model starts from the contact. One can see a quite representative example of imperfect fitting of the experimental data with the Hertz model (a similar observation is true for the Sneddon model). The brush model requires fitting the region of maximum load, in which brush is almost completely squeezed. It is typical to see rather good fitting of the force curve with the brush model. Fig. 2 b demonstrates the force due to the cellular brush derived with Eq. 5. One can clearly see the exponential dependence (straight line) when showing the force in the logarithmic scale. The scope of this work is to analyze the self-consistency of the models based on the behavior of the elastic modulus. Therefore, we will focus on the self-consistency of calculations of the elastic modulus.

Figure 2.

An example of processing raw data, deflection of the AFM cantilever versus vertical position of the AFM scanner (d versus Z) collected with a spherical AFM indenter (approaching curve). (a) The regions of the curve fitted with the Hertz and brush model are shown. The fitting region in the Hertz model starts from the contact, whereas the region for the brush model is near the maximum load where brush is squeezed. (b) Fitting the brush force curve with the steric brush model (solid line); see Eq. 6. To see this figure in color, go online.

Note that only the approaching part of the force-indentation curves has been typically analyzed. Here we follow the same approach. The approaching force curve is chosen for two reasons:

• 1.The brush may have insufficient time to relax and give undisturbed contribution to the force curves.
• 2.The cell, being an alive and active object, may react to the indenting by quick restructuring of its cytoskeleton (which recovers very quickly, e.g., Zhao et al. (38)), altering the retracting curve.

Although this alteration seems to be small when a dull indenter is used (25), it is safer to use the approaching part of the curve. In addition, one can obtain the information about undisturbed brush when using the approaching curve.

We now describe an important modification of the brush models used here compared to those published previously in the literature (15,31,47,54). When using the brush model in a self-consistent way, one can sometimes obtain the elastic modulus for a rather narrow range of indentation depths. This is because the methodology of the brush model relies on the assumption that the brush is reasonably compressed near the maximum load (see the description in Materials and Methods). On the one hand, one can increase the maximum load force to extend the range of possible indentation depths. Such an approach, however, can have two problems: The AFM probe can 1), start indenting internal cell organelles, which are typically more rigid, as well as 2), start detecting the rigid substrate. This usually results in a sharp increase of the elastic modulus with increasing indentation.

For specific purposes of this work, it is instructive to compare the elastic modulus derived for small indentations (which can easily be done using the Hertz and Sneddon models). Here we describe a simple recursion approach, which allows deriving the elastic modulus for smaller indentations in the brush models. To derive the elastic modulus for smaller indentations in which the brush is not completely squeezed, one can use the following logic:

After deriving the elastic modulus for large indentations (near the maximum of the load force when the brush is completely squeezed), and calculating the brush parameters by using Eqs. 6 or 11, one can recursively use the brush parameters in Eqs. 5 or 9, which describe deformations of (not completely squeezed) brush and cell body simultaneously. Now Eqs. 5 or 9 can be treated as the equations to fit the experimental data with respect to just one unknown variable, the elastic modulus. It is plausible to keep the second parameter Z₀ (position of nondeformed cell surface) fixed at the value derived for the large indentations because it is physically impossible that the cell has multiple nondeformed boundaries simultaneously. As previously, the fit should be done within the applicability of the brush equations (Eqs. 6 or 11), i.e., 0.1 < h/L < 0.8. The modulus derived from Eqs. 4 or 8 is associated with substantially smaller forces/indentations than the ones derived for a completely squeezed brush. Thus, the elastic modulus can be derived for the extended region of indentation depths. If needed, this range could be extended even more if one uses a slightly more accurate model of the brush, using the power law rather than the exponential form (49).

Fig. 3 demonstrates the elastic modulus of the cells derived with the brush models described above. Fitting the modulus for small parts of different fitting regions, we can test the dependence of the elastic modulus on the indentation depth (defined as the maximum of the fitting region). One can see that the elastic modulus derived for the conical indenter (Fig. 3 a) still shows a strong dependence on the indentation depth. Most of the time, the modulus is decreasing with the increase of the indentation depth. At the same time, the modulus derived for the spherical probe (Fig. 3 b) shows almost no dependence on the indentation depth. Starting from some indentation, nevertheless, the modulus may demonstrate some increase. This is expected because the AFM probe eventually starts squeezing the cell organelles, in particular, the nucleus (because in the above examples, the cell is deformed right above the nucleus). Furthermore, for higher indentations, the AFM probe can obviously start detecting the contribution of the rigid substrate. The cell height used in this work is ∼15 μm (estimated from confocal microscopy, see, e.g., Iyer et al. (15)). Depth independence of the modulus is typically observed until ∼1.5 μm, 10% of the cell height.

Figure 3.

Brush models, with representative examples of the dependence of the elastic modulus (shown in kPa) on the indentation depth (shown in nanometers) of human cervical epithelial cells when using (a) conical and (b) spherical AFM indenters. To see this figure in color, go online.

It is interesting to note that a steep decrease of the elastic modulus with the indentation depth derived for the sharp conical probe (Fig. 3 a) looks very close to the case of indenting soft polymers with a similar conical AFM probe (22,55). Similarly, much higher values of the modulus were observed with the sharp indenter compared to the case of the dull probe. Moreover, it was found that the elastic modulus became indentation-independent when a sufficiently dull probe was used. In those works, such behavior was found to be a result of nonlinear stress-strain relation induced by an excessively sharp conical indenter. It was found by comparing the estimate stresses under the sharp indenter with the limits of linearity measured directly for a macroscopical block of the same polymeric material. It is, however, impossible to grow a macroscopic-size cell. Although it is plausible to conceive that the sharp conical probe induces nonlinear stress-strain relation (see also the support of this idea in Dimitriadis et al. (56) and Costa and Yin (57)), the situation may be more complicated (see, e.g., Vargas-Pinto et al. (58), in which this effect was explained by the presence of a cell cortex layer, which was shown for endothelial cells).

Similar calculations done using classical Hertz (for a spherical indenter) and Sneddon (for a sharp conical indenter) models are exemplified in Fig. 4. One can see no constancy of the elastic modulus when changing the indentation depth. Again, here we observe the similarity with the case of indenting soft polymers with conical and dull AFM probes (22,55). The case of the sharp probe demonstrated the decrease of the modulus with indentation (due to nonlinearity in stress-strain relation), whereas the case of the dull probe showed the increase of the modulus with the indentation increase due to squeezing out nano-asperities (due to roughness) of the polymer surface. Those nano-asperities are presumably analogous to the brush in the case of cells.

Figure 4.

The elastic modulus derived in the Sneddon/Hertz models. Representative examples of the dependence of the elastic modulus (shown in kPa) on the indentation depth (shown in nanometers) of human cervical epithelial cells when using conical (a, left) and spherical (b, right) AFM probes (note: brush not taken into account). To see this figure in color, go online.

As we noted previously, the physical meaning of the indentation depth in the models with and without brush are rather different. The models with brush consider the deformation of the cell body only, whereas the Hertz and Sneddon models measure the total deformation of both the brush and cell body. Therefore, when comparing the dependence of the modulus on indentation, it is instructive to plot the dependence of the elastic modulus on the load force. The load force is a model-independent alternative of the indentation depth. Fig. 5 shows such comparison done for the most interesting case of the spherical indenter. One can clearly see that the practical independence of the elastic modulus on the load force derived in the brush model is in clear contrast with the dependence observed in the Hertz model.

Figure 5.

Comparison of the elastic modulus on the load force derived in (a) the Hertz model and (b) the brush model (the same raw data was used for the both models). To see this figure in color, go online.

To confirm the observed behavior for a statistically sound number of cells and indentation depths, 30–40 cancer and normal cells were used for the indenting with either sharp or dull probes. To begin, we present the elastic modulus for different indentations and cells with the help of two-dimensional histograms used previously in Fuhrmann et al. (59). The frequency of occurrence of a specific value of the modulus for a specific indentation depth is color-encoded. Fig. 6 shows the histograms of distributions of the elastic modulus of all normal and cancer cells derived for different indentations (forces) in different models. Although these histograms are a good representation of the all data analyzed, the histograms can be quite confusing to interpret. To help to compare with the response of an ideal elastic material, we added two bottom panels to Fig. 6. Those panels show theoretical histograms generated for a normal distribution of the elastic modulus of an ideal elastic homogeneous/isotropic material under the same conditions as in the experiment (the same range of the indentation depths). Two theoretical histograms correspond to normal and cancer cells of the study. The average modulus and standard deviation in the models were taken equal to the average and standard deviation of the modulus obtained for all (either normal or cancer) cells. Qualitative comparison of the histograms for the model and the cell data shows that similarity between the brush model and ideal elastic material is better compared to the results of the Hertz model.

Figure 6.

The case of the spherical indenter. (a–f) Histograms of distributions of the elastic modulus of normal and cancer cells derived for various indentations in different models. (c and f) Histograms showing behavior of an ideal elastic material under the same conditions. To see this figure in color, go online.

Results of a similar study, but done for conical sharp indenters, are shown in Fig. 7. Although one can still see a better qualitative agreement between ideal elastic material and the brush model, the agreement is less pronounced compared to the case of the spherical indenter. Furthermore, compared to Figs. 3 and 4, the histograms can give only a rather qualitative result. For example, it does not allow seeing the difference in strong indentation dependence in the case of the sharp conical indenter and brush model.

Figure 7.

The case of the conical indenter. (a–f) Histograms of distributions of the elastic modulus of normal and cancer cells derived for different indentations in different models. (c and f) Histograms showing behavior of ideal elastic materials under the same conditions. To see this figure in color, go online.

To understand how we can present the statistical data to confirm the independence of the modulus, let us discuss how these histograms were created in more detail. At first glance, the histograms derived with the help of the brush model (Fig. 6, middle row) may even seem to contradict the independence of the elastic modulus of the indentation depth demonstrated in Fig. 3 b. However, the histogram shows a statistical distribution of multiple moduli/cells. The modulus measured for each single point/cell may be indeed independent of the indentation depth. At the same time, AFM operates with a range of forces on cells with different elastic properties. Thus, more-rigid cells will present less indentation depth range, and vice versa. When plotted in one histogram, this hides the dependence of the elastic modulus of the indentation depth. The anatomy of these histograms is exemplified in Fig. 8. Each line shows separate modulus-indentation dependence at each point of a cell. The ideal material gives no indentation dependence, whereas cancer cells indented with a cone indenter demonstrate strong indentation dependence. One can see how those dependences form the histograms of Figs. 6 and 7.

Figure 8.

The schematics of the histograms of Figs. 6 and 7 showing the actual depth dependence for the modulus measurement at a single surface point. Examples of cancer cells are shown: (a) for the case of Fig. 7f (an ideal elastic material, cone indenter, Sneddon model); (b) for the case of Fig. 7d (cone indenter, Sneddon model); (c) for the case of Fig. 6d (spherical indenter, Hertz model); and (d) for the case of Fig. 6e (spherical indenter, brush model). To see this figure in color, go online.

To analyze the dependence of the elastic modulus on the indentation depth both statistically and quantitatively, we introduce the statistical distribution of gradients of the modulus. Mathematically, it is easy to find as a standard deviation of the elastic modulus St.DEV(E) calculated for the each modulus-indentation curve normalized by the average modulus Aver(E):

$$St.Dev\left(E\right)/Aver\left(E\right)\times 100\%.$$ (12)

Because such a definition will depend on how many point/values of the modulus for different indentations we consider, we kept the equal number of indentations for all cells and models (taken within approximately the same range of 0.2–3.5 μm for different cells and models). These values are calculated for different locations in different cells. Table 1 shows the results for normal and cancer cells indented with sharp and dull AFM indenters. Approximately 300 values were calculated for each cell type for each indenter. The smaller this number, the less the modulus is dependent on the indentation depth. One can clearly see that the brush model applied to the indentation data obtained with the dull probe shows the smallest gradients. For reference, the value of the elastic modulus of the cells used in this study was found to be 1.60 ± 0.60 kPa (normal cells) and 1.40 ± 0.48 kPa (cancer cells).

Table 1.

Comparison of variability of the values of the elastic modulus E over the cell surface within one cell and within 20 cells

Values measured	20 cells		1 cell
	Sphere probe	Cone probe	Sphere probe	Cone probe
E, kPa	1.60 ± 0.60	6.6 ± 2.1	1.59 ± 0.40	7.1 ± 3.5

It is interesting to compare variability of the values of the elastic modulus over the cell surface within one cell with variability between cells. The example of 20 cells versus one representative cell is shown in Table 2. The important conclusion here is that the surface variability within one cell is comparable or even higher (typical for the cone probe) than intercell variations. This stresses the importance of statistical measurements rather than just one point per cell. While the results for normal cells are shown in Table 1, similar results can be found for the cancer cells.

Table 2.

Results of compilation of the elastic modulus dependency on the indentation depth for normal and cancer cells indented with sharp and dull AFM indenters

Spherical probe
Normal cells		Cancer cells
Hertz	Brush	Hertz	Brush
41%	6%	40%	5%
Sharp (conical) probe
Normal cells		Cancer cells
Sneddon	Brush	Sneddon	Brush
47%	31%	43%	35%

Approximately 300 values were calculated for each cell type for each indenter.

It should be noted that Eq. 12 describes the gradient of the modulus quite well only in the case of monotonic/smooth dependence of the modulus on the indentation. If the values of the modulus were excessively noisy, Eq. 12 would give a large number for flat but noisy modulus dependence. Examples shown in Figs. 3–5 demonstrated that the dependences observed are relatively smooth, and therefore, we can use Eq. 11 as the measure of modulus dependency on indentation. Obviously, this approach is not unique. However, we conceive that it statistically proves our conclusion about independence of the elastic modulus derived in the brush model on the data collected with the dull AFM probe.

There are two main reasons for the above observation:

• 1.The contribution of the cellular brush to the cell mechanics apparently cannot be ignored. Even for the cone probe, which, being sharp, may effectively penetrate through the brush more easily than the spherical probe, it develops a rather large area of contact sufficient to detect the brush. Taking the brush into account is paramount, as the brush shows an essentially nonelastic response to the load force.
• 2.The other reason is presumably related to the excessive stresses/strains induced by the sharp probe, which leads to a strong modulus-indentation dependence for small indentations. This was not observed when a dull AFM probe was used.

Conclusion

Here we demonstrated that cells can be described with the elastic modulus (effective Young’s modulus) in a self-consistent way if the pericellular brush layer is taken into account as a separate cellular structure. The brush layer is essentially a nonelastic part of a cell that is better described as an entropic/steric brush. The rest of the cell (cell body) can be described with just one value of the elastic modulus (virtually indentation-independent if indenting below 10% of the cell height). This is a rather nontrivial result because cells are highly heterogeneous objects.

The above conclusion was reached after performing a comparative analysis of the elastic modulus derived from the AFM indentation data obtained on human cervical epithelial cells (both normal and cancerous) using both sharp conical and dull spherical AFM probes. The indentation data were processed through different elastic models, in which the cells body was considered either coated or not with the pericellular brush (microvilli, microridges, glycocalyx, etc.). Although the presence of the pericellular brush is known, the reason we considered the models with no brush was to answer the question whether the brush contribution to the cell mechanics is significant. Furthermore, the majority of the models used so far did not take the cellular brush into account, and therefore, such models should be considered for comparison.

Independence of the elastic modulus of the indentation depth is the necessary condition of applicability of any model in which the material is considered linearly elastic and homogeneous/isotropic. Such independence was demonstrated here to exist only for the elastic modulus derived when using the brush model on the data collected with the dull AFM probe. These observations lead to the conclusion that it is possible to describe the elastic properties of the cell body with the elastic modulus after separating the contribution of the cellular brush. A sharp conical probe brings strong modulus-indentation dependence for small indentations. This is presumably due to the excessively high stresses/strains produced by this sharp indenter, a phenomenon observed when indenting polymers.