Probing mechanical properties of living cells by atomic force microscopy with blunted pyramidal cantilever tips

Atomic force microscopy (AFM) allows the acquisition of high-resolution images and the measurement of mechanical properties of living cells under physiological conditions. AFM cantilevers with blunted pyramidal tips are commonly used to obtain images of living cells. Measurement of mechanical properties with these tips requires a contact model that takes into account their blunted geometry. The aim of this work was to develop a contact model of a blunted pyramidal tip and to assess the suitability of pyramidal tips for probing mechanical properties of soft gels and living cells. We developed a contact model of a blunted pyramidal tip indenting an elastic half-space. We measured Young’s modulus $\left(E\right)$ and the complex shear modulus $(G^{*}=G^{\prime }+\mathrm{i}{G}^{″})$ of agarose gels and A549 alveolar epithelial cells with pyramidal tips and compared them with those obtained with spherical tips. The gels exhibited an elastic behavior with almost coincident loading and unloading force curves and negligible values of $G^{″}$. $E$ fell sharply with indentation up to $\sim 300\mathrm{nm}$, showing a linear regime for deeper indentations. A similar indentation dependence of $E$ with twofold lower values at the linear regime was obtained with the spherical tip fitted with Hertz’s model. The dependence of $E$ on indentation in cells paralleled that found in gels. Cells exhibited viscoelastic behavior with $G^{″}∕G^{\prime }\sim 1∕4$. Pyramidal tips commonly used for AFM imaging are suitable for probing mechanical properties of soft gels and living cells.

Figure 1

(Color online) Model of the blunted pyramidal AFM tip. (a) Section of a blunted pyramidal tip with semi-included angle $\theta$, tip defect $h$, and spherical cap radius $R_c$ indenting an elastic half-space. $F$ is force, $\delta$ is indentation, $b$ is the radial distance corresponding to the transition from spherical cap to pyramidal faces, and $h^{*}=b^2∕2R_c$. (b) Three-dimensional plot of the end $\left(1\mu \mathrm{m}\right)$ of the blunted four-sided pyramidal tip model with the spherical cap merging tangential with the pyramid faces ($\theta =35°$, $R_c=100\mathrm{nm}$). Dashed line is the contour corresponding to flat faces (ideal pyramid).

Figure 2

Scanning electron micrographs of (a) pyramidal and (b) spherical tips used in agarose gels $(\mathrm{bar}=1\mu \mathrm{m})$. The inset shows the blunted pyramidal approximation $(\mathrm{bar}=100\mathrm{nm})$.

Figure 3

Examples of deflection-distance $\left(d\text{−}z\right)$ curves measured on approach (dotted lines) and retraction (dashed lines) on agarose gels with (a) pyramidal and (b) spherical tips. Solid lines are the corresponding blunted pyramidal and spherical fits to approach curves. Fitted values were $E=0.76\mathrm{kPa}$, $z_c=634\mathrm{nm}$, and $d_{\mathit{off}}=-196\mathrm{nm}$, and $E=0.35\mathrm{kPa}$, $z_c=1651\mathrm{nm}$, and $d_{\mathit{off}}=-207\mathrm{nm}$, respectively. Arrows indicate the contact point. Insets display residual errors.

Figure 4

Indentation dependence $(\text{mean}\pm \mathrm{SE})$ of Young’s modulus of agarose gels $(n=7)$ obtained with the (a) pyramidal and (b) spherical tips. Dashed line in (a) displays data fitted with the ideal pyramid model. Dotted lines are mean $E$ values obtained by fitting the whole $d\text{−}z$ curves.

Figure 5

(a) Height (left) and deflection (right) images of an A549 cell obtained with the pyramidal tip in contact mode (scan $\text{size}=50\mu \mathrm{m}$). Gray level ranges $6\mu \mathrm{m}$ (left) and $50\mathrm{nm}$ (right). (b) and (c) Examples of deflection-distance $\left(d\text{−}z\right)$ curves measured on approach (dotted lines) and retraction (dashed lines) on the cell with (b) pyramidal and (c) spherical tips. Fitted values were $E=1.31\mathrm{kPa}$, $z_c=1997\mathrm{nm}$, and $d_{\mathit{off}}=-49\mathrm{nm}$, and $E=0.36\mathrm{kPa}$, $z_c=1812\mathrm{nm}$, and $d_{\mathit{off}}=-51\mathrm{nm}$, respectively. Arrows indicate the contact point. Insets display residual errors. The crosses in (a) indicate the points where the pyramidal force curve was obtained.

Figure 6

Indentation dependence $(\text{mean}\pm \mathrm{SE})$ of Young’s modulus of alveolar epithelial cells $(n=7)$ obtained with the (a) pyramidal and (b) spherical tips. Dashed line in (a) displays data fitted with the ideal pyramid model. Dotted lines are mean $E$ values obtained by fitting the whole $d\text{−}z$ curves.

Figure 7

Force-indentation $\left(F\text{−}\delta \right)$ relationship for blunted pyramidal, ideal pyramidal, and spherical models on an elastic half-space with Young’s modulus $E=1\mathrm{kPa}$. Solid line is the blunted pyramidal model with semi-included angle $\theta =35°$ and a spherical cap of radius $R_c=100\mathrm{nm}$ merging tangential with the faces. Dotted line is the ideal pyramidal model with $\theta =35°(\text{slope}=2)$. Dashed line is the spherical model of radius $R=100\mathrm{nm}(\text{slope}=3∕2)$. Dashed-dotted line is the spherical model with $R=2500\mathrm{nm}(\text{slope}=3∕2)$. The blunted pyramidal model matches the spherical model of $R=100\mathrm{nm}(R_c=R)$ for indentations smaller than $80\mathrm{nm}(\delta <2h^{*})$ and approaches the ideal pyramidal model for deeper indentations. The difference is $<15\%$ for indentations deeper than $1000\mathrm{nm}$.