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Quantitative Study of the Elastic Modulus of Loosely Attached Cells in AFM Indentation Experiments
Biophysical Journal Volume 104 May 2013 2123–2131
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Quantitative Study of the Elastic Modulus of Loosely Attached Cells in AFM Indentation Experiments Maxim E. Dokukin,†§ Nataliia V. Guz,§ and Igor Sokolov†‡§{* † §
Department of Mechanical Engineering and ‡Department of Biomedical Engineering, Tufts University, Medford, Massachusetts; and Department of Physics and {Nanoengineering and Biotechnology Laboratories Center, Clarkson University, Potsdam, New York
ABSTRACT When measuring the elastic (Young’s) modulus of cells using AFM, good attachment of cells to a substrate is paramount. However, many cells cannot be firmly attached to many substrates. A loosely attached cell is more compliant under indenting. It may result in artificially low elastic modulus when analyzed with the elasticity models assuming firm attachment. Here we suggest an AFM-based method/model that can be applied to extract the correct Young’s modulus of cells loosely attached to a substrate. The method is verified by using primary breast epithelial cancer cells (MCF-7) at passage 4. At this passage, approximately one-half of cells develop enough adhesion with the substrate to be firmly attached to the substrate. These cells look well spread. The other one-half of cells do not develop sufficient adhesion, and are loosely attached to the substrate. These cells look spherical. When processing the AFM indentation data, a straightforward use of the Hertz model results in a substantial difference of the Young’s modulus between these two types of cells. If we use the model presented here, we see no statistical difference between the values of the Young’s modulus of both poorly attached (round) and firmly attached (close to flat) cells. In addition, the presented model allows obtaining parameters of the brush surrounding the cells. The cellular brush observed is also statistically identical for both types of cells. The method described here can be applied to study mechanics of many other types of cells loosely attached to substrates, e.g., blood cells, some stem cells, cancerous cells, etc.
INTRODUCTION Over the last decade, many studies have demonstrated the link between mechanics of human cells and various diseases and abnormalities, such as cancer (1–6), arthritis (7), malaria (8), ischemia (9), and even aging (10–12). Most types of cells in the human body, like muscle, epithelial, blood cells, neurons, etc., stay under a permanently changing force environment. The changes in cell mechanics may change the mechanical response of tissue or organs. It is plausible, therefore, to expect that the alteration of cell mechanics may lead to various pathologies or diseases. Thus, the analysis of basic parameters of cell mechanics is an important instrument to obtain new fundamental insights into diseases, to help the development of new methods of diagnosis. It is important to develop methods that allow measuring the elastic parameters independently of the method and instrument used. The Young’s modulus is one of such characteristics. AFM is a convenient method to study soft materials (13). AFM can work as a microscopy tool for imaging of cells (14–16). Due to its unique capability to detect forces between a probe and sample, it has been widely used not only for imaging but also to measure the various physical properties of cells (4), in particular, cell mechanics (1–3,10,17–23). The Hertz model (24) and its various modifications (2,20,25) have been widely used to determine the elasticity, i.e., the Young’s modulus of cells. In these models, the cell Submitted August 26, 2012, and accepted for publication April 4, 2013. *Correspondence: [email protected]
is assumed to be a homogenous material elastic material with a flat boundary; the cell shape is often not taken into account. While effective homogeneity of the cell material may be considered as a reasonable approximation for small deformations, the cell surface is typically far from being flat. Various membrane protrusions can be detectable with AFM (26). A typical eukaryotic cell is surrounded with a brush of molecular components of the plasma membrane (glycocalyx) as well as protrusions of the membrane itself in the form of microvilli, microridges, cilia, or filopodia (27,28). This brush layer is responsible for the cell-cell interaction (29), cell migration (30), differentiation, and proliferation (31,32). The brush is important during embryonic development (33), in wound healing (34), inflammation (35,36), and mammalian fertilization (37). It is involved in epithelialmesenchymal transition (33), resistance to apoptosis, and multidrug resistance (38). Molecular entropic brushes are known to surround neurofilaments to maintain interfilament spacing (39,40). It has been recently found that the cellular brush would interfere with indentation measurements of elastic properties of cell body, and a new model had to be used (41). The model separates the contribution of the cellular brush and the deformation of the cell body. Interestingly, cancer cells may look softer than normal if the cellular brush is not taken into account as was shown in the case of human cervical epithelial cells (1). The next assumption of the models used is that the cell is firmly adhered to the substrate. However, cells do not necessarily develop a strong adhesion to substrates. If a cell does not adhere firmly to a substrate, it is easier to deform compared to a firmly attached one. Consequently,
Editor: Michael Edidin. Ó 2013 by the Biophysical Society 0006-3495/13/05/2123/9 $2.00
http://dx.doi.org/10.1016/j.bpj.2013.04.019
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a straightforward use of the elasticity models to analyze the indentation of loosely attached cells while assuming firm attachment will result in an artificially low elastic modulus. To avoid this artifact, a creative method has been developed to attach such cells by mechanically immobilizing cells in microfabricated wells (42). This, however, requires a special substrate, which may potentially change the cell mechanics because of additional stresses imposed upon the cell due to the mechanical immobilization. Here we present a model that allows using AFM to derive a correct value of the Young’s modulus of cells weakly attached to substrates. We verify the model on breast epithelial cancer MCF-7 cells of small passage (passage 4). At this passage, approximately one-half of the cell population develops sufficiently strong adhesion to the substrate; the cells are firmly attached to the substrate. The other half of the cell population is loosely attached to the substrate. We demonstrate that a straightforward use of the Hertz model results in a substantial error in the value of the Young’s modulus of the cells. If we use the model suggested in this article, which takes into account the existence of a cellular brush (1,41), we see no statistical difference between the values of the Young’s modulus for both loosely attached (spherical) and firmly attached (well-spread or semispherical) cells. The requirement of strong attachment of cells to the substrate restricts the number of possible cell studies. For example, many blood cells, especially cancer cells in metastasis, cannot naturally develop a sufficiently strong adhesion to a majority of substrates. Moreover, there is always a risk of altering the cell properties when trying to modify the substrate to enhance the adhesion. It is well known that mechanical properties of cells can be influenced by the substrate; see, e.g., Tee et al. (43). Therefore, the results presented in this work will help to extend the study of the cell mechanics by means of the AFM technique to a broader class of cells that are poorly attached to the substrate. Finally, the results presented in this work demonstrate, one more time, the importance of taking into account the cellular brush when studying cell mechanics with indentation methods.
METHODS Models Here we describe the models used in this work to derive the Young’s modulus of the cell body as well as the parameters of the surrounding brush. These models will be compared with the standard Hertz model, which is briefly overviewed below as well.
Deformation of firmly attached cells The cell is assumed to be a dome-shaped elastic medium; for example, a semispherical shape firmly attached to a nondeformable substrate, covered with uniform brush, as shown in Fig.1 a. The major parts of this model have already been described in Iyer et al. (1) and Sokolov et al. (41), in which a Biophysical Journal 104(10) 2123–2131
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FIGURE 1 Schematics of the interaction between an AFM spherical indenter (probe) and (a) firmly and (b) loosely attached cells. Deformations of both the cell body and surrounding cellular brush are shown. Z is the relative position of the cantilever; d is the cantilever deflection; Z0 is nondeformed position of the cell body; i, itop, and ibottom are the deformations of the cell body; h and htop are the separation distances between the cell body and the probe; and hbottom is the distance between the cell body and substrate.
flat cell was considered. Because we will deal with cells that are smaller than considered previously, a finite radius of the cell has to be taken into account. Here we briefly outline this model by focusing on the finite radius of the cell. Because the adhesion between the indenter and surface is negligible compared to the indenting force, the Hertz contact model (24) is used to describe the deformation of the elastic part of the cell-brush system, in what we call the cell body. Fig. 1 a shows a scheme of the interaction of a spherical AFM probe/indenter with a cell covered with a brush. The parameters presented in the scheme are related as (41)
Z ¼ Z0 i d þ h;
(1)
where Z is the relative vertical position of the AFM scanner, Z0 is the vertical position for a nondeformed cell body, i is the deformation of the cell
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body, and h is the probe-cell body distance. The position Z ¼ 0 was chosen when the cantilever deflection d is maximum. It is assumed that the cell’s brush is completely squeezed (h ¼ 0) near the maximum indentation force (i.e., maximum deflection). This allows using the Hertz model to calculate the Young’s modulus E of the cell body from the experimentally measured parameters d and i,
"
9 k i ¼ Z0 d Z ¼ d2=3 $ 16 E
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #2=3 Rprobe þ Rcell ; Rprobe Rcell
(2)
(3)
To describe the brush parameters quantitatively, the following equation describing steric interaction between a spherical probe of radius Rprobe and a semispherical cell of radius Rcell due to the existence of an entropic brush can be used (1,41,44) as
FðhÞz50 kB T R N 3=2 exp 2p
" i ¼ d2=3
Fðh Þ ¼ kd ¼ k$ðh Z þ Z0 i Þ;
Deformation of loosely attached cells Here we develop a model which allows us to find the Young’s modulus E of the cell body in the case of cells loosely attached to a rigid substrate. We assume that the cell is attached through the interaction between the brush and substrate, and such interaction does not alter the spherical shape of the cell. This assumption is verified with confocal optical microscopy (see Results and Discussion). Thus, when deforming with an AFM indenter, the brush is squeezed between both cell-probe and cell-substrate interfaces. Fig. 1 b graphically describes such a situation. The presented parameters can now be connected as
(8)
where h* ¼ htopþ hbottom. To find the parameters of the cell brush, we assume that we are dealing with an entropic steric brush (1,41,44). This is justified by a good fit of the experimental data with this model (see Results and Discussion). The force due to such a brush can be described as
F htop ¼ Fðhbottom Þ
htop exp 2p ¼ L1 hbottom 3=2 ¼ 50kB TR2 N2 L2 exp 2p L2 3=2 kdz50kB TR1 N1 L1
(4)
where kB is the Boltzmann constant, T is the temperature, R* ¼ Rprobe $ Rcell/Rprobe þ Rcell, N is the surface density of the brush constituents (grafting density, effective molecular density), and L is the thickness of the brush layer.
Z ¼ Z0 itop ibottom d þ htop þ hbottom ;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2=3 2=3 # Rprobe þ Rcell 9 k pffiffiffiffiffiffiffiffi þ : Rprobe Rcell 16 E Rcell
Here the Poisson ratio is v ¼ 0.5 (same as in the previous model). Using Eq. 5 and known fitting parameters i* and Z0, we can now find the force-separation curve F(h), which corresponds to the deformation of the brush on both sides of the cell. It is given by the formula
h L; L
9 k 16 E
(7)
where k is the spring constant of the AFM cantilever, and Rprobe and Rcell indicate the radii of the AFM probe and cell body, respectively. In the case of a flat cell, Rcell tends to infinity. The Poisson ratio v of a cell is typically chosen to be 0.5 (because of a small range of possible variations of v, the error due to the uncertainty its definition is small and can be ignored). The force F(h) caused by the existence of the brush can be reconstructed using Eq. 1 and the Young’s modulus calculated as described above:
FðhÞ ¼ kd ¼ k$ðh Z þ Z0 iÞ:
static loading, the force acting on both sides of the cell is equal to the load force F ¼ kd. Therefore, the total deformation can be written as
(9)
where kB is the Boltzmann constant; T is temperature; R* ¼ Rprobe , Rcell/ Rprobe þ Rcell is the effective cell-probe radius; R2 ¼ Rcell is cell radius; N1 and N2 are the effective surface density of the brush constituents such as grafting density or effective molecular surface density of the brush; and L1 and L2 are the sizes of the top and bottom brushes, respectively. Equation 9 can be rewritten as
2pL2 M 3=2 Fðh Þz50 kB T R1 N1 exp L1 þ L2 2ph L1 ; exp L1 þ L2
(10)
where
! 3=2 1 50 kB T R1 N1 L1 ln M ¼ : 3=2 2p 50 kB T R2 N2 L2
(5)
where Z is the relative vertical scanner position, and Z ¼ 0 is chosen when d ¼ dmax. The value Z0 is the scanner position for the nondeformed position of the cell body, htop is the separation distance between the cell body and probe, hbottom is the distance between the cell body and substrate, and itop and ibottom are the top and bottom deformations of the cell body, respectively. To decouple the brush contribution, we assume that the brush on the top and bottom of the cell can be completely squeezed (hbottomþ htop ¼ 0) starting from some force. For such deformations, one can rewrite Eq. 5 as
From Eq. 10, one can see that it is impossible to extract separate parameters of the top and bottom brushes. Furthermore, Eq. 10 can be reduced to a simple single steric brush equation provided we assume that the brush lengths of L1 and L2 are equal. Introducing an effective brushplength ffiffiffiffiffiffiffiffiffiffiffi L* (L1 ¼ L2 ¼ L*/2) and an effective grafting density N ¼ N1 N2, Eq. 10 can be rewritten as
Z ¼ Z0 i d;
pffiffiffiffiffiffiffiffiffiffi where R ¼ 1=2 R1 R2 is the effective radius. This reduced form of the brush description allows us to extract the effective parameters of both top and bottom brushes unambiguously.
(6)
where i* ¼ itopþ ibottom is the total deformation of the cell body. Parameters Z and d are directly measured in the AFM experiment when collecting the force-indentation curves. The other parameters Z0 and i* can be found from the fitting of the force-indentation curves. As in the previous model, the Hertz equations are used to describe the total deformation of the cell body (the sum of the top and bottom deformations). Due to the quasi-
h exp 2p L ; L
Fðh Þz50 kB T R N
3=2
(11)
The Hertz model Because we will compare the models described above with the Hertz model widely used in the literature (10,20,25,45–47), we briefly describe the Hertz Biophysical Journal 104(10) 2123–2131
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model as well. In that approach, the cell is treated as a homogenous smooth (well-defined boundary) medium of radius Rcell. To derive the Young’s modulus of the cell, the experimental force-indentation curves are fitted with the formula
16 E Fðic Þ ¼ 9
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rprobe $Rcell 3=2 i ; Rprobe þ Rcell c
(12)
where F is the load force, E is the Young’s modulus, and ic is the indentation (or sometime called penetration) depth. Here the Poisson ratio of the cells is set to 0.5.
Experimental Cells Cancer MCF-7 cell line was provided by American Type Culture Collection (ATCC, Manassas, VA). Cells were cultured in 60-mm dishes with ATCCformulated Eagle’s Minimum Essential Medium (Cat. No. 30-2003) with 0.01 mg/mL bovine insulin and 10% bovine serum in an incubator (at 5% CO2 and 37 C temperature) according to the ATCC protocol. The medium was changed three times per week. Cells were studied when at passage 4. Before measurements, cells were placed on a sterilized (in 70% ethanol for 30 min) microscope glass cover (GaineXpress, Hong Kong) in the same medium but without serum and kept for 16 h. Right before the experiment, all samples were gently rinsed with the serum-free medium to remove possibly detached cells, and left for 30 min in the same serum-free medium in the incubator.
Atomic force microscopy Bioscope Catalyst (Bruker/Veeco, Santa Barbara, CA) AFM placed on a U2000 confocal Eclipse C1 microscope (Nikon, Melville, NY) and Dimension 3100 (Bruker/Veeco) AFM with Nanoscope V controller and NPoint close-loop scanner (200 mm 200 mm 30 mm, XYZ) were used in this study. Standard cantilever holders for operation in liquids were employed. To obtain the distribution of the properties over the cell surface and simultaneously record cell topography, the force-volume mode of operation was utilized. The force curves were collected with the vertical ramp size of 5–6 mm. To minimize viscoelastic effects, force-indentation curves were recorded with a frequency of 1 Hz. The physical speed (mm/s) used is about two times higher than what is typically used in the literature because we used almost double larger ramp size. This is because the time of indentation is preferable to keep unchanged due to cell creep/probe deepening under constant load and cell movement. At the same time, the ramp size used is about twice larger than the previously used due to the need in larger deformation of weakly adhered cells (deformed from the top and the bottom simultaneously). The force-volume images of cells were collected with the resolution of 16 16 pixels (typically within 50 50 mm2 area).
we take the force curves in the surface points around the top when the incline of the surface is <10–15 . See the Supporting Material for detail. 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). On the next step, force curves from the identified cell areas were processed through the models described above. A nonlinear curve fitting of corresponding equations allows deriving both the Young’s modulus of the cell body and parameters of the brush (length and grafting density). Based on the optical microscope images, all cells were preliminarily separated into two groups: well spread (firmly attached) and spherical (loosely attached). On average, 10 force-indentation curves per cell were analyzed.
RESULTS AND DISCUSSION After plating on sterilized glass slides, MCF-7 cells do not demonstrate good adhesion in the first passages. Passage 4 was identified as an intermediate one, in which approximately one-half of the cells developed enough adhesion with the substrate to be firmly attached to it. These cells look well spread through an optical microscope. The other half of the cells do not develop sufficient adhesion, and they are loosely attached to the substrate. These cells look spherical. Representative optical images are shown in Fig. 2. An inset in Fig. 2 demonstrates a confocal side view of both well-spread and spherical-looking cells. This justifies our assumption: we will treat the flat-looking cells as firmly attached semispherical objects (Fig. 1 a), whereas the spherical-looking cells will be considered as loosely attached spheres (Fig. 1 b). The presence of the cellular brush can be qualitatively seen in the confocal images. However, a quantitative characterization of the brush can only be done through the analysis of the force curves, as described in the Experimental section. Following the method of Iyer et al. (1) and Sokolov et al. (41), we assumed that starting from some load force such as cantilever deflection, the surrounding brush can be completely squeezed. With those forces, the AFM probe deforms the cell body directly. This part of the cell deformation curve should then be described by Eqs. 2 and 6,
AFM probe: spherical indenter Standard V-shaped 200-mm AFM tipless cantilevers (Bruker/Veeco) were used throughout the study. Five-mm-diameter silica balls (Bangs Laboratories, Fishers, IN) were glued to the cantilevers as described in Berdyyeva et al. (10) and Ong and Sokolov (48). The radius of the probe was measured by imaging the inverse grid (Cat. No. TGT1; NT-NGT, Russia). The cantilever spring constant was measured using the thermal tuning method before gluing the spherical probe. The spring constant of the cantilevers used in this work was found to be 0.069 N/m.
Data processing method The models described in Methods were developed for a known geometry as a sphere over either plane or hemisphere. Thus, we processed only the force curves from the top area of the cell; following the previous works (1,41,49), Biophysical Journal 104(10) 2123–2131
FIGURE 2 Representative optical images of firmly (well-spread images) and loosely (spherical images) adhered cells. (Right panel, inset) Confocal side view of both well-spread and spherical-looking cells. Confocal images are not to scale (the heights of cells in the confocal images are between 10 and 20 mm).
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which is the de facto Hertz model applied to the cell body with no brush. It should be noted that the Hertz model can be applied to a homogeneous material only. This is a rather nontrivial assumption of mechanical homogeneity of a highly biologically inhomogeneous cell body. Furthermore, when the stress-strain relation is no longer linear, even a homogeneous material can be overstretched in the indentation experiments. In such a case, the concept of modulus of elasticity will no longer be applicable. Here we assumed that 1. The cell body can be considered as a homogeneous material for small deformations, and 2. We are dealing with the linear stress-strain response. These are two typical assumptions of virtually all publications on this topic. However, here we are checking these two assumptions by testing both a posteriori. Specifically, we test the strong condition of linearity of homogeneous material (50): the elastic modulus should be independent of the indentation depth. Having either inhomogeneity or nonlinearity will break the modulus independence of the indentation depth. We checked this dependence for all cells measured in this work. Typical examples of modulus-depth dependence for firmly and weakly adhered cells will be shown later (Figs. 3 b and 4 b). It should be stressed that these assumptions are not necessarily true for all cells, forces, and probes. For example, the force required to squeeze the brush may be excessively high. The probe may start squeezing some organelles inside the cell when the brush is sufficiently squeezed. As a result, the cell body cannot be described as a linear elastic medium. The Young’s modulus calculated for such a case should increase with the probe penetration (an example is shown in Fig. 3 b). Therefore, we would suggest that testing the independence of the Young’s modulus of the indentation depth should be used that the part of the measurement protocol. Fig. 3 a exemplifies this approach for the case of firmly adhered cells. For the curve presented in Fig. 3 a, elastic deformation of the cell body is well fitted starting from the cantilever deflection d ¼ 60–100 nm (the load force of 4.1–6.9 nN). Below these forces, the fitted curve substan-
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tially deviates from the experimental data. This implies that we have simultaneous deformation of both the brush and cell’s body. Above those forces, we can calculate the Young’s modulus of the cell body for different probe penetrations/indentations, defined as the middle indentation point of the fitting region. The results are shown in Fig. 3 b. One can see that the Young’s modulus is rather independent of probe penetration. Nevertheless, starting from some penetrations, the modulus rapidly increases. This is expected because the AFM probe eventually starts squeezing the cell organelles and the nucleus in particular, because the cell is deformed right above the nucleus. To find the parameters of the brush surrounding cells, the force-separation curve corresponding due to the brush was first calculated through Eq. 3. The brush parameters, length (L) and grafting density (N), were derived from the forceseparation curve by fitting with Eq. 4 for the region 0.2 < h/L < 0.9 (44). Both the force-separation curve and fitting are shown in Fig. 3 c. The same procedure was repeated for loosely attached (spherical) cells (see Fig. 1 b). Representative curves are shown in Fig. 4. One can see that these are similar to those of the firmly attached cells. Fig. 5 shows histograms of the distributions of the Young’s modulus for both firmly and loosely attached cells. A total of 15 firmly and 15 loosely attached cells in the culture dish were analyzed. Fig. 5 shows the Young’s moduli calculated using the brush model for firmly adhered cells (Fig. 5 a) and the brush model for loosely attached cells (Fig. 5 b). One can see that the difference between the elastic moduli of the firmly and loosely attached cells is negligible within their variability: 0.95 5 0.26 kPa vs. 1.19 5 0.38 kPa for the firmly and loosely attached cells, respectively. From a practical point of view, this difference is also negligible because the error in the measurement of the spring constant of the AFM cantilever can reach 15– 20% (51). To compare our models with the widely used simple Hertz model (that is, when no brush is taken into account), we processed experimental raw data through the Hertz model shown in Eq. 12. A representative example of such
FIGURE 3 A representative example of the processing of the force-indentation curves for the case of firmly adhered cells. (a) Raw data and partial fitting of the curve with Eqs. 1 and 2. The fitting parameters were Z0 and the Young’s modulus; (b) dependence of the derived Young’s modulus on the indentation depth; and (c) fitting of the cellular brush with Eq. 4 (done and shown for 0.2 < h/L < 0.9). Biophysical Journal 104(10) 2123–2131
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FIGURE 4 A representative example of the processing of the force-indentation curves for the case of loosely adhered cells. (a) Raw data and partial fitting of the curve with Eqs. 6 and 7. The fitting parameters were Z0 and the Young’s modulus; (b) dependence of the derived Young’s modulus on indentation; and (c) fitting of the cellular brush with Eq. 11 (done and shown for 0.2 < h/L < 0.9).
processing is shown in Fig. 6. One can see from Fig. 6 that it is impossible to fit well the entire range of the experimental data. We conceive that this is due to the presence of the cellular brush, which cannot be described as an elastic material. The dependence of the Young’s modulus on the indentation depth is shown in Fig. 6 c. One can see that it is heavily indentation-dependent. This is a violation of the necessary condition of stress-strain linearity condition (see, e.g., Tang et al. (50)), which implies that the concept of the elastic modulus could not be used at all when such a behavior is observed. For the purpose of comparison with many previous works we will ignore this violation, and will use the initial part of the curve to derive the Young’s modulus (small cantilever deflection d, or forces) in which the deviation from the Hertz model is small.
Fig. 5, c and d, shows the distributions of the Young’s modulus derived for the firmly and loosely attached cells, respectively. The Hertz model is applied here. The average moduli are 0.40 5 0.30 kPa for the firmly attached cells (it is rather close to the values previously reported in Li et al. (52)) and 0.21 0.17 kPa for the loosely attached cells. One can see that the Young’s moduli for both types of cells are different by approximately a factor of two. This can be explained by the fact that the AFM probe deforms mostly the brush at the small deformations. The loosely attached cells have the double brush, i.e., the brush on the top and the bottom of the cells, whereas the firmly attached cells have the brush only on the top of the cells (as could be seen in Fig. 2). When the AFM probe deforms a loosely attached cell, both the top and bottom are compressed,
FIGURE 5 The distribution of the Young’s modulus calculated using the brush models for (a) firmly and (b) loosely attached cells, and using Hertz model for (c) firmly and (d) loosely attached cells.
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FIGURE 6 A representative example of processing of raw force data (Z versus d) with the Hertz model (Eq. 12). Scattered points represent the raw experimental data. (a) Fitting on the entire range of forces. (b) Fitting of the initial part of the force curve (small deflections/forces; see dashed line for the mathematical extrapolation outside the fitting region). (c) Dependence of the Young’s modulus derived with the Hertz model on the indentation depth.
and therefore, the effective Young’s modulus should be twice less compared to the deformation of the single brush. An important feature of the brush methods used here is the ability to extract information not only about rigidity of cell body but also quantitative information about the cellular brush. Fig. 7 shows the brush parameters, length L and effective grafting density N derived with the entropic brush model. One can see from Fig. 7 that both firmly and loosely attached cells demonstrate similar brush length parameters. The average value of brush length L is 1020 5 600 nm for firmly attached cells and L ¼ 1250 5 900 nm for loosely attached cells. (Note that we assumed that the brush lengths L of the top and bottom brushes are equal for the loosely attached cells, as derived from Eq. 11.) The average grafting density N is 300 5 210 mm2 and 230 5 230 mm2 for the
firmly and loosely attached cells, respectively. A relatively large SD comes from a noticeable variability between individual cells. To amplify, the brush models result in rather similar values of the Young’s modulus and brush parameters for both firmly and loosely attached cells observed within passage 4 of MCF-7 cells. It might be plausible but not obvious to expect that the cells grown under the same physiological conditions should have similar biophysical characteristics, including rigidity and brush. It may happen because we measure the cell indentation right above the nucleus. The cell adhesion causes reorganization of cytoskeletal, F-actin. At the same time, it was found that during adhesion the actin reorganization mostly appears near the periphery of a cell (47,53). One can expect that the area
FIGURE 7 Distributions of the brush length (L) and grafting density (N) are shown for (a and c) firmly attached and (b and d) loosely attached cells, respectively.
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above the nucleus would not change significantly. Therefore, the observed similarity might be considered in favor of the models proposed. In contrast, the results obtained with the Hertz model show a twofold difference between firmly and loosely attached cells. The values derived with the help of the Hertz model are from three-to-five times smaller than Young’s modulus calculated using the brush models. The Young’s modulus derived through the Hertz model is strongly indentation-dependent, which implies that this model cannot be used to extract the modulus of elasticity at all. In addition, it is worth mentioning that in the opposite to the brush models, the moduli distributions derived with the Hertz model are highly nonGaussian, with most probably shifted to smaller values (Fig. 5, c and d). CONCLUSIONS Here we suggested a model to derive the mechanical properties of cells both loosely and firmly attached to a rigid substrate from the indentation data obtained by means of AFM. The model takes into account the deformations of the cell body and the brush that surrounds the cell. The derived parameters include the Young’s modulus, the length, and the grafting density of the cellular brush. The model was verified on MCF-7 cells of a relatively small passage (passage 4) in which approximately onehalf of cells are firmly attached to the substrate while the other half is loosely attached. We demonstrated that the Young’s modulus of both loosely and firmly attached cells show virtually the same values within 1 SD. Moreover, the brush parameters, brush length, and grafting density are also similar for both loosely and firmly attached cells (the difference is only in the physical distribution of the brush: the loosely attached cells are completely surrounded by the brush). We also showed that if one uses the standard Hertz model with no brush taken into account, a twofold difference in the Young’s modulus was observed between firmly and loosely attached cells. This artifact could easily be explained by the presence of the brush. The values of the Young’s modulus derived with the Hertz model were from three-to-five times smaller than Young’s modulus calculated using the brush models. This is a clear artifact due to the deformation of a relatively soft brush that is misinterpreted, and the deformation of the cell body. It is importantly to note that the Young’s modulus derived in the Hertz model shows a strong dependence on the indentation depth, which violates the necessary condition of stress-strain linearity, and consequently, it does not allow the model to be used. In contrast, the brush models show a clear plateau, a region in which the Young’s modulus is depth-independent. The method described can be applied to study the mechanics of many other types of cells which are either difficult to attach to a substrate (e.g., some stem and Biophysical Journal 104(10) 2123–2131
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cancerous cells) or the attachment can substantially change cells (e.g., red blood cells on glass, or cells on substrates treated with poly-L-lysine). SUPPORTING MATERIAL Details of the methods and error analysis are available at http://www. biophysj.org/biophysj/supplemental/S0006-3495(13)00449-9. We thank Dr. Craig Woodworth (Clarkson University) for the help with the growth and storage of cells. American Type Culture Collection (ATCC) is acknowledged for providing MCF-7 cells.
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Quantitative Study of the Elastic Modulus of Loosely Attached Cells in AFM Indentation Experiments Maxim E. Dokukin,†§ Nataliia V. Guz,§ and Igor Sokolov†‡§{* † §
Department of Mechanical Engineering and ‡Department of Biomedical Engineering, Tufts University, Medford, Massachusetts; and Department of Physics and {Nanoengineering and Biotechnology Laboratories Center, Clarkson University, Potsdam, New York
ABSTRACT When measuring the elastic (Young’s) modulus of cells using AFM, good attachment of cells to a substrate is paramount. However, many cells cannot be firmly attached to many substrates. A loosely attached cell is more compliant under indenting. It may result in artificially low elastic modulus when analyzed with the elasticity models assuming firm attachment. Here we suggest an AFM-based method/model that can be applied to extract the correct Young’s modulus of cells loosely attached to a substrate. The method is verified by using primary breast epithelial cancer cells (MCF-7) at passage 4. At this passage, approximately one-half of cells develop enough adhesion with the substrate to be firmly attached to the substrate. These cells look well spread. The other one-half of cells do not develop sufficient adhesion, and are loosely attached to the substrate. These cells look spherical. When processing the AFM indentation data, a straightforward use of the Hertz model results in a substantial difference of the Young’s modulus between these two types of cells. If we use the model presented here, we see no statistical difference between the values of the Young’s modulus of both poorly attached (round) and firmly attached (close to flat) cells. In addition, the presented model allows obtaining parameters of the brush surrounding the cells. The cellular brush observed is also statistically identical for both types of cells. The method described here can be applied to study mechanics of many other types of cells loosely attached to substrates, e.g., blood cells, some stem cells, cancerous cells, etc.
INTRODUCTION Over the last decade, many studies have demonstrated the link between mechanics of human cells and various diseases and abnormalities, such as cancer (1–6), arthritis (7), malaria (8), ischemia (9), and even aging (10–12). Most types of cells in the human body, like muscle, epithelial, blood cells, neurons, etc., stay under a permanently changing force environment. The changes in cell mechanics may change the mechanical response of tissue or organs. It is plausible, therefore, to expect that the alteration of cell mechanics may lead to various pathologies or diseases. Thus, the analysis of basic parameters of cell mechanics is an important instrument to obtain new fundamental insights into diseases, to help the development of new methods of diagnosis. It is important to develop methods that allow measuring the elastic parameters independently of the method and instrument used. The Young’s modulus is one of such characteristics. AFM is a convenient method to study soft materials (13). AFM can work as a microscopy tool for imaging of cells (14–16). Due to its unique capability to detect forces between a probe and sample, it has been widely used not only for imaging but also to measure the various physical properties of cells (4), in particular, cell mechanics (1–3,10,17–23). The Hertz model (24) and its various modifications (2,20,25) have been widely used to determine the elasticity, i.e., the Young’s modulus of cells. In these models, the cell Submitted August 26, 2012, and accepted for publication April 4, 2013. *Correspondence: [email protected]
is assumed to be a homogenous material elastic material with a flat boundary; the cell shape is often not taken into account. While effective homogeneity of the cell material may be considered as a reasonable approximation for small deformations, the cell surface is typically far from being flat. Various membrane protrusions can be detectable with AFM (26). A typical eukaryotic cell is surrounded with a brush of molecular components of the plasma membrane (glycocalyx) as well as protrusions of the membrane itself in the form of microvilli, microridges, cilia, or filopodia (27,28). This brush layer is responsible for the cell-cell interaction (29), cell migration (30), differentiation, and proliferation (31,32). The brush is important during embryonic development (33), in wound healing (34), inflammation (35,36), and mammalian fertilization (37). It is involved in epithelialmesenchymal transition (33), resistance to apoptosis, and multidrug resistance (38). Molecular entropic brushes are known to surround neurofilaments to maintain interfilament spacing (39,40). It has been recently found that the cellular brush would interfere with indentation measurements of elastic properties of cell body, and a new model had to be used (41). The model separates the contribution of the cellular brush and the deformation of the cell body. Interestingly, cancer cells may look softer than normal if the cellular brush is not taken into account as was shown in the case of human cervical epithelial cells (1). The next assumption of the models used is that the cell is firmly adhered to the substrate. However, cells do not necessarily develop a strong adhesion to substrates. If a cell does not adhere firmly to a substrate, it is easier to deform compared to a firmly attached one. Consequently,
Editor: Michael Edidin. Ó 2013 by the Biophysical Society 0006-3495/13/05/2123/9 $2.00
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a straightforward use of the elasticity models to analyze the indentation of loosely attached cells while assuming firm attachment will result in an artificially low elastic modulus. To avoid this artifact, a creative method has been developed to attach such cells by mechanically immobilizing cells in microfabricated wells (42). This, however, requires a special substrate, which may potentially change the cell mechanics because of additional stresses imposed upon the cell due to the mechanical immobilization. Here we present a model that allows using AFM to derive a correct value of the Young’s modulus of cells weakly attached to substrates. We verify the model on breast epithelial cancer MCF-7 cells of small passage (passage 4). At this passage, approximately one-half of the cell population develops sufficiently strong adhesion to the substrate; the cells are firmly attached to the substrate. The other half of the cell population is loosely attached to the substrate. We demonstrate that a straightforward use of the Hertz model results in a substantial error in the value of the Young’s modulus of the cells. If we use the model suggested in this article, which takes into account the existence of a cellular brush (1,41), we see no statistical difference between the values of the Young’s modulus for both loosely attached (spherical) and firmly attached (well-spread or semispherical) cells. The requirement of strong attachment of cells to the substrate restricts the number of possible cell studies. For example, many blood cells, especially cancer cells in metastasis, cannot naturally develop a sufficiently strong adhesion to a majority of substrates. Moreover, there is always a risk of altering the cell properties when trying to modify the substrate to enhance the adhesion. It is well known that mechanical properties of cells can be influenced by the substrate; see, e.g., Tee et al. (43). Therefore, the results presented in this work will help to extend the study of the cell mechanics by means of the AFM technique to a broader class of cells that are poorly attached to the substrate. Finally, the results presented in this work demonstrate, one more time, the importance of taking into account the cellular brush when studying cell mechanics with indentation methods.
METHODS Models Here we describe the models used in this work to derive the Young’s modulus of the cell body as well as the parameters of the surrounding brush. These models will be compared with the standard Hertz model, which is briefly overviewed below as well.
Deformation of firmly attached cells The cell is assumed to be a dome-shaped elastic medium; for example, a semispherical shape firmly attached to a nondeformable substrate, covered with uniform brush, as shown in Fig.1 a. The major parts of this model have already been described in Iyer et al. (1) and Sokolov et al. (41), in which a Biophysical Journal 104(10) 2123–2131
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FIGURE 1 Schematics of the interaction between an AFM spherical indenter (probe) and (a) firmly and (b) loosely attached cells. Deformations of both the cell body and surrounding cellular brush are shown. Z is the relative position of the cantilever; d is the cantilever deflection; Z0 is nondeformed position of the cell body; i, itop, and ibottom are the deformations of the cell body; h and htop are the separation distances between the cell body and the probe; and hbottom is the distance between the cell body and substrate.
flat cell was considered. Because we will deal with cells that are smaller than considered previously, a finite radius of the cell has to be taken into account. Here we briefly outline this model by focusing on the finite radius of the cell. Because the adhesion between the indenter and surface is negligible compared to the indenting force, the Hertz contact model (24) is used to describe the deformation of the elastic part of the cell-brush system, in what we call the cell body. Fig. 1 a shows a scheme of the interaction of a spherical AFM probe/indenter with a cell covered with a brush. The parameters presented in the scheme are related as (41)
Z ¼ Z0 i d þ h;
(1)
where Z is the relative vertical position of the AFM scanner, Z0 is the vertical position for a nondeformed cell body, i is the deformation of the cell
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body, and h is the probe-cell body distance. The position Z ¼ 0 was chosen when the cantilever deflection d is maximum. It is assumed that the cell’s brush is completely squeezed (h ¼ 0) near the maximum indentation force (i.e., maximum deflection). This allows using the Hertz model to calculate the Young’s modulus E of the cell body from the experimentally measured parameters d and i,
"
9 k i ¼ Z0 d Z ¼ d2=3 $ 16 E
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #2=3 Rprobe þ Rcell ; Rprobe Rcell
(2)
(3)
To describe the brush parameters quantitatively, the following equation describing steric interaction between a spherical probe of radius Rprobe and a semispherical cell of radius Rcell due to the existence of an entropic brush can be used (1,41,44) as
FðhÞz50 kB T R N 3=2 exp 2p
" i ¼ d2=3
Fðh Þ ¼ kd ¼ k$ðh Z þ Z0 i Þ;
Deformation of loosely attached cells Here we develop a model which allows us to find the Young’s modulus E of the cell body in the case of cells loosely attached to a rigid substrate. We assume that the cell is attached through the interaction between the brush and substrate, and such interaction does not alter the spherical shape of the cell. This assumption is verified with confocal optical microscopy (see Results and Discussion). Thus, when deforming with an AFM indenter, the brush is squeezed between both cell-probe and cell-substrate interfaces. Fig. 1 b graphically describes such a situation. The presented parameters can now be connected as
(8)
where h* ¼ htopþ hbottom. To find the parameters of the cell brush, we assume that we are dealing with an entropic steric brush (1,41,44). This is justified by a good fit of the experimental data with this model (see Results and Discussion). The force due to such a brush can be described as
F htop ¼ Fðhbottom Þ
htop exp 2p ¼ L1 hbottom 3=2 ¼ 50kB TR2 N2 L2 exp 2p L2 3=2 kdz50kB TR1 N1 L1
(4)
where kB is the Boltzmann constant, T is the temperature, R* ¼ Rprobe $ Rcell/Rprobe þ Rcell, N is the surface density of the brush constituents (grafting density, effective molecular density), and L is the thickness of the brush layer.
Z ¼ Z0 itop ibottom d þ htop þ hbottom ;
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!2=3 2=3 # Rprobe þ Rcell 9 k pffiffiffiffiffiffiffiffi þ : Rprobe Rcell 16 E Rcell
Here the Poisson ratio is v ¼ 0.5 (same as in the previous model). Using Eq. 5 and known fitting parameters i* and Z0, we can now find the force-separation curve F(h), which corresponds to the deformation of the brush on both sides of the cell. It is given by the formula
h L; L
9 k 16 E
(7)
where k is the spring constant of the AFM cantilever, and Rprobe and Rcell indicate the radii of the AFM probe and cell body, respectively. In the case of a flat cell, Rcell tends to infinity. The Poisson ratio v of a cell is typically chosen to be 0.5 (because of a small range of possible variations of v, the error due to the uncertainty its definition is small and can be ignored). The force F(h) caused by the existence of the brush can be reconstructed using Eq. 1 and the Young’s modulus calculated as described above:
FðhÞ ¼ kd ¼ k$ðh Z þ Z0 iÞ:
static loading, the force acting on both sides of the cell is equal to the load force F ¼ kd. Therefore, the total deformation can be written as
(9)
where kB is the Boltzmann constant; T is temperature; R* ¼ Rprobe , Rcell/ Rprobe þ Rcell is the effective cell-probe radius; R2 ¼ Rcell is cell radius; N1 and N2 are the effective surface density of the brush constituents such as grafting density or effective molecular surface density of the brush; and L1 and L2 are the sizes of the top and bottom brushes, respectively. Equation 9 can be rewritten as
2pL2 M 3=2 Fðh Þz50 kB T R1 N1 exp L1 þ L2 2ph L1 ; exp L1 þ L2
(10)
where
! 3=2 1 50 kB T R1 N1 L1 ln M ¼ : 3=2 2p 50 kB T R2 N2 L2
(5)
where Z is the relative vertical scanner position, and Z ¼ 0 is chosen when d ¼ dmax. The value Z0 is the scanner position for the nondeformed position of the cell body, htop is the separation distance between the cell body and probe, hbottom is the distance between the cell body and substrate, and itop and ibottom are the top and bottom deformations of the cell body, respectively. To decouple the brush contribution, we assume that the brush on the top and bottom of the cell can be completely squeezed (hbottomþ htop ¼ 0) starting from some force. For such deformations, one can rewrite Eq. 5 as
From Eq. 10, one can see that it is impossible to extract separate parameters of the top and bottom brushes. Furthermore, Eq. 10 can be reduced to a simple single steric brush equation provided we assume that the brush lengths of L1 and L2 are equal. Introducing an effective brushplength ffiffiffiffiffiffiffiffiffiffiffi L* (L1 ¼ L2 ¼ L*/2) and an effective grafting density N ¼ N1 N2, Eq. 10 can be rewritten as
Z ¼ Z0 i d;
pffiffiffiffiffiffiffiffiffiffi where R ¼ 1=2 R1 R2 is the effective radius. This reduced form of the brush description allows us to extract the effective parameters of both top and bottom brushes unambiguously.
(6)
where i* ¼ itopþ ibottom is the total deformation of the cell body. Parameters Z and d are directly measured in the AFM experiment when collecting the force-indentation curves. The other parameters Z0 and i* can be found from the fitting of the force-indentation curves. As in the previous model, the Hertz equations are used to describe the total deformation of the cell body (the sum of the top and bottom deformations). Due to the quasi-
h exp 2p L ; L
Fðh Þz50 kB T R N
3=2
(11)
The Hertz model Because we will compare the models described above with the Hertz model widely used in the literature (10,20,25,45–47), we briefly describe the Hertz Biophysical Journal 104(10) 2123–2131
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model as well. In that approach, the cell is treated as a homogenous smooth (well-defined boundary) medium of radius Rcell. To derive the Young’s modulus of the cell, the experimental force-indentation curves are fitted with the formula
16 E Fðic Þ ¼ 9
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rprobe $Rcell 3=2 i ; Rprobe þ Rcell c
(12)
where F is the load force, E is the Young’s modulus, and ic is the indentation (or sometime called penetration) depth. Here the Poisson ratio of the cells is set to 0.5.
Experimental Cells Cancer MCF-7 cell line was provided by American Type Culture Collection (ATCC, Manassas, VA). Cells were cultured in 60-mm dishes with ATCCformulated Eagle’s Minimum Essential Medium (Cat. No. 30-2003) with 0.01 mg/mL bovine insulin and 10% bovine serum in an incubator (at 5% CO2 and 37 C temperature) according to the ATCC protocol. The medium was changed three times per week. Cells were studied when at passage 4. Before measurements, cells were placed on a sterilized (in 70% ethanol for 30 min) microscope glass cover (GaineXpress, Hong Kong) in the same medium but without serum and kept for 16 h. Right before the experiment, all samples were gently rinsed with the serum-free medium to remove possibly detached cells, and left for 30 min in the same serum-free medium in the incubator.
Atomic force microscopy Bioscope Catalyst (Bruker/Veeco, Santa Barbara, CA) AFM placed on a U2000 confocal Eclipse C1 microscope (Nikon, Melville, NY) and Dimension 3100 (Bruker/Veeco) AFM with Nanoscope V controller and NPoint close-loop scanner (200 mm 200 mm 30 mm, XYZ) were used in this study. Standard cantilever holders for operation in liquids were employed. To obtain the distribution of the properties over the cell surface and simultaneously record cell topography, the force-volume mode of operation was utilized. The force curves were collected with the vertical ramp size of 5–6 mm. To minimize viscoelastic effects, force-indentation curves were recorded with a frequency of 1 Hz. The physical speed (mm/s) used is about two times higher than what is typically used in the literature because we used almost double larger ramp size. This is because the time of indentation is preferable to keep unchanged due to cell creep/probe deepening under constant load and cell movement. At the same time, the ramp size used is about twice larger than the previously used due to the need in larger deformation of weakly adhered cells (deformed from the top and the bottom simultaneously). The force-volume images of cells were collected with the resolution of 16 16 pixels (typically within 50 50 mm2 area).
we take the force curves in the surface points around the top when the incline of the surface is <10–15 . See the Supporting Material for detail. 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). On the next step, force curves from the identified cell areas were processed through the models described above. A nonlinear curve fitting of corresponding equations allows deriving both the Young’s modulus of the cell body and parameters of the brush (length and grafting density). Based on the optical microscope images, all cells were preliminarily separated into two groups: well spread (firmly attached) and spherical (loosely attached). On average, 10 force-indentation curves per cell were analyzed.
RESULTS AND DISCUSSION After plating on sterilized glass slides, MCF-7 cells do not demonstrate good adhesion in the first passages. Passage 4 was identified as an intermediate one, in which approximately one-half of the cells developed enough adhesion with the substrate to be firmly attached to it. These cells look well spread through an optical microscope. The other half of the cells do not develop sufficient adhesion, and they are loosely attached to the substrate. These cells look spherical. Representative optical images are shown in Fig. 2. An inset in Fig. 2 demonstrates a confocal side view of both well-spread and spherical-looking cells. This justifies our assumption: we will treat the flat-looking cells as firmly attached semispherical objects (Fig. 1 a), whereas the spherical-looking cells will be considered as loosely attached spheres (Fig. 1 b). The presence of the cellular brush can be qualitatively seen in the confocal images. However, a quantitative characterization of the brush can only be done through the analysis of the force curves, as described in the Experimental section. Following the method of Iyer et al. (1) and Sokolov et al. (41), we assumed that starting from some load force such as cantilever deflection, the surrounding brush can be completely squeezed. With those forces, the AFM probe deforms the cell body directly. This part of the cell deformation curve should then be described by Eqs. 2 and 6,
AFM probe: spherical indenter Standard V-shaped 200-mm AFM tipless cantilevers (Bruker/Veeco) were used throughout the study. Five-mm-diameter silica balls (Bangs Laboratories, Fishers, IN) were glued to the cantilevers as described in Berdyyeva et al. (10) and Ong and Sokolov (48). The radius of the probe was measured by imaging the inverse grid (Cat. No. TGT1; NT-NGT, Russia). The cantilever spring constant was measured using the thermal tuning method before gluing the spherical probe. The spring constant of the cantilevers used in this work was found to be 0.069 N/m.
Data processing method The models described in Methods were developed for a known geometry as a sphere over either plane or hemisphere. Thus, we processed only the force curves from the top area of the cell; following the previous works (1,41,49), Biophysical Journal 104(10) 2123–2131
FIGURE 2 Representative optical images of firmly (well-spread images) and loosely (spherical images) adhered cells. (Right panel, inset) Confocal side view of both well-spread and spherical-looking cells. Confocal images are not to scale (the heights of cells in the confocal images are between 10 and 20 mm).
Elastic Modulus of Loosely Attached Cells
which is the de facto Hertz model applied to the cell body with no brush. It should be noted that the Hertz model can be applied to a homogeneous material only. This is a rather nontrivial assumption of mechanical homogeneity of a highly biologically inhomogeneous cell body. Furthermore, when the stress-strain relation is no longer linear, even a homogeneous material can be overstretched in the indentation experiments. In such a case, the concept of modulus of elasticity will no longer be applicable. Here we assumed that 1. The cell body can be considered as a homogeneous material for small deformations, and 2. We are dealing with the linear stress-strain response. These are two typical assumptions of virtually all publications on this topic. However, here we are checking these two assumptions by testing both a posteriori. Specifically, we test the strong condition of linearity of homogeneous material (50): the elastic modulus should be independent of the indentation depth. Having either inhomogeneity or nonlinearity will break the modulus independence of the indentation depth. We checked this dependence for all cells measured in this work. Typical examples of modulus-depth dependence for firmly and weakly adhered cells will be shown later (Figs. 3 b and 4 b). It should be stressed that these assumptions are not necessarily true for all cells, forces, and probes. For example, the force required to squeeze the brush may be excessively high. The probe may start squeezing some organelles inside the cell when the brush is sufficiently squeezed. As a result, the cell body cannot be described as a linear elastic medium. The Young’s modulus calculated for such a case should increase with the probe penetration (an example is shown in Fig. 3 b). Therefore, we would suggest that testing the independence of the Young’s modulus of the indentation depth should be used that the part of the measurement protocol. Fig. 3 a exemplifies this approach for the case of firmly adhered cells. For the curve presented in Fig. 3 a, elastic deformation of the cell body is well fitted starting from the cantilever deflection d ¼ 60–100 nm (the load force of 4.1–6.9 nN). Below these forces, the fitted curve substan-
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tially deviates from the experimental data. This implies that we have simultaneous deformation of both the brush and cell’s body. Above those forces, we can calculate the Young’s modulus of the cell body for different probe penetrations/indentations, defined as the middle indentation point of the fitting region. The results are shown in Fig. 3 b. One can see that the Young’s modulus is rather independent of probe penetration. Nevertheless, starting from some penetrations, the modulus rapidly increases. This is expected because the AFM probe eventually starts squeezing the cell organelles and the nucleus in particular, because the cell is deformed right above the nucleus. To find the parameters of the brush surrounding cells, the force-separation curve corresponding due to the brush was first calculated through Eq. 3. The brush parameters, length (L) and grafting density (N), were derived from the forceseparation curve by fitting with Eq. 4 for the region 0.2 < h/L < 0.9 (44). Both the force-separation curve and fitting are shown in Fig. 3 c. The same procedure was repeated for loosely attached (spherical) cells (see Fig. 1 b). Representative curves are shown in Fig. 4. One can see that these are similar to those of the firmly attached cells. Fig. 5 shows histograms of the distributions of the Young’s modulus for both firmly and loosely attached cells. A total of 15 firmly and 15 loosely attached cells in the culture dish were analyzed. Fig. 5 shows the Young’s moduli calculated using the brush model for firmly adhered cells (Fig. 5 a) and the brush model for loosely attached cells (Fig. 5 b). One can see that the difference between the elastic moduli of the firmly and loosely attached cells is negligible within their variability: 0.95 5 0.26 kPa vs. 1.19 5 0.38 kPa for the firmly and loosely attached cells, respectively. From a practical point of view, this difference is also negligible because the error in the measurement of the spring constant of the AFM cantilever can reach 15– 20% (51). To compare our models with the widely used simple Hertz model (that is, when no brush is taken into account), we processed experimental raw data through the Hertz model shown in Eq. 12. A representative example of such
FIGURE 3 A representative example of the processing of the force-indentation curves for the case of firmly adhered cells. (a) Raw data and partial fitting of the curve with Eqs. 1 and 2. The fitting parameters were Z0 and the Young’s modulus; (b) dependence of the derived Young’s modulus on the indentation depth; and (c) fitting of the cellular brush with Eq. 4 (done and shown for 0.2 < h/L < 0.9). Biophysical Journal 104(10) 2123–2131
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FIGURE 4 A representative example of the processing of the force-indentation curves for the case of loosely adhered cells. (a) Raw data and partial fitting of the curve with Eqs. 6 and 7. The fitting parameters were Z0 and the Young’s modulus; (b) dependence of the derived Young’s modulus on indentation; and (c) fitting of the cellular brush with Eq. 11 (done and shown for 0.2 < h/L < 0.9).
processing is shown in Fig. 6. One can see from Fig. 6 that it is impossible to fit well the entire range of the experimental data. We conceive that this is due to the presence of the cellular brush, which cannot be described as an elastic material. The dependence of the Young’s modulus on the indentation depth is shown in Fig. 6 c. One can see that it is heavily indentation-dependent. This is a violation of the necessary condition of stress-strain linearity condition (see, e.g., Tang et al. (50)), which implies that the concept of the elastic modulus could not be used at all when such a behavior is observed. For the purpose of comparison with many previous works we will ignore this violation, and will use the initial part of the curve to derive the Young’s modulus (small cantilever deflection d, or forces) in which the deviation from the Hertz model is small.
Fig. 5, c and d, shows the distributions of the Young’s modulus derived for the firmly and loosely attached cells, respectively. The Hertz model is applied here. The average moduli are 0.40 5 0.30 kPa for the firmly attached cells (it is rather close to the values previously reported in Li et al. (52)) and 0.21 0.17 kPa for the loosely attached cells. One can see that the Young’s moduli for both types of cells are different by approximately a factor of two. This can be explained by the fact that the AFM probe deforms mostly the brush at the small deformations. The loosely attached cells have the double brush, i.e., the brush on the top and the bottom of the cells, whereas the firmly attached cells have the brush only on the top of the cells (as could be seen in Fig. 2). When the AFM probe deforms a loosely attached cell, both the top and bottom are compressed,
FIGURE 5 The distribution of the Young’s modulus calculated using the brush models for (a) firmly and (b) loosely attached cells, and using Hertz model for (c) firmly and (d) loosely attached cells.
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FIGURE 6 A representative example of processing of raw force data (Z versus d) with the Hertz model (Eq. 12). Scattered points represent the raw experimental data. (a) Fitting on the entire range of forces. (b) Fitting of the initial part of the force curve (small deflections/forces; see dashed line for the mathematical extrapolation outside the fitting region). (c) Dependence of the Young’s modulus derived with the Hertz model on the indentation depth.
and therefore, the effective Young’s modulus should be twice less compared to the deformation of the single brush. An important feature of the brush methods used here is the ability to extract information not only about rigidity of cell body but also quantitative information about the cellular brush. Fig. 7 shows the brush parameters, length L and effective grafting density N derived with the entropic brush model. One can see from Fig. 7 that both firmly and loosely attached cells demonstrate similar brush length parameters. The average value of brush length L is 1020 5 600 nm for firmly attached cells and L ¼ 1250 5 900 nm for loosely attached cells. (Note that we assumed that the brush lengths L of the top and bottom brushes are equal for the loosely attached cells, as derived from Eq. 11.) The average grafting density N is 300 5 210 mm2 and 230 5 230 mm2 for the
firmly and loosely attached cells, respectively. A relatively large SD comes from a noticeable variability between individual cells. To amplify, the brush models result in rather similar values of the Young’s modulus and brush parameters for both firmly and loosely attached cells observed within passage 4 of MCF-7 cells. It might be plausible but not obvious to expect that the cells grown under the same physiological conditions should have similar biophysical characteristics, including rigidity and brush. It may happen because we measure the cell indentation right above the nucleus. The cell adhesion causes reorganization of cytoskeletal, F-actin. At the same time, it was found that during adhesion the actin reorganization mostly appears near the periphery of a cell (47,53). One can expect that the area
FIGURE 7 Distributions of the brush length (L) and grafting density (N) are shown for (a and c) firmly attached and (b and d) loosely attached cells, respectively.
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above the nucleus would not change significantly. Therefore, the observed similarity might be considered in favor of the models proposed. In contrast, the results obtained with the Hertz model show a twofold difference between firmly and loosely attached cells. The values derived with the help of the Hertz model are from three-to-five times smaller than Young’s modulus calculated using the brush models. The Young’s modulus derived through the Hertz model is strongly indentation-dependent, which implies that this model cannot be used to extract the modulus of elasticity at all. In addition, it is worth mentioning that in the opposite to the brush models, the moduli distributions derived with the Hertz model are highly nonGaussian, with most probably shifted to smaller values (Fig. 5, c and d). CONCLUSIONS Here we suggested a model to derive the mechanical properties of cells both loosely and firmly attached to a rigid substrate from the indentation data obtained by means of AFM. The model takes into account the deformations of the cell body and the brush that surrounds the cell. The derived parameters include the Young’s modulus, the length, and the grafting density of the cellular brush. The model was verified on MCF-7 cells of a relatively small passage (passage 4) in which approximately onehalf of cells are firmly attached to the substrate while the other half is loosely attached. We demonstrated that the Young’s modulus of both loosely and firmly attached cells show virtually the same values within 1 SD. Moreover, the brush parameters, brush length, and grafting density are also similar for both loosely and firmly attached cells (the difference is only in the physical distribution of the brush: the loosely attached cells are completely surrounded by the brush). We also showed that if one uses the standard Hertz model with no brush taken into account, a twofold difference in the Young’s modulus was observed between firmly and loosely attached cells. This artifact could easily be explained by the presence of the brush. The values of the Young’s modulus derived with the Hertz model were from three-to-five times smaller than Young’s modulus calculated using the brush models. This is a clear artifact due to the deformation of a relatively soft brush that is misinterpreted, and the deformation of the cell body. It is importantly to note that the Young’s modulus derived in the Hertz model shows a strong dependence on the indentation depth, which violates the necessary condition of stress-strain linearity, and consequently, it does not allow the model to be used. In contrast, the brush models show a clear plateau, a region in which the Young’s modulus is depth-independent. The method described can be applied to study the mechanics of many other types of cells which are either difficult to attach to a substrate (e.g., some stem and Biophysical Journal 104(10) 2123–2131
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cancerous cells) or the attachment can substantially change cells (e.g., red blood cells on glass, or cells on substrates treated with poly-L-lysine). SUPPORTING MATERIAL Details of the methods and error analysis are available at http://www. biophysj.org/biophysj/supplemental/S0006-3495(13)00449-9. We thank Dr. Craig Woodworth (Clarkson University) for the help with the growth and storage of cells. American Type Culture Collection (ATCC) is acknowledged for providing MCF-7 cells.
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