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TECHNICAL PAPERS: Cell

# Non-Hertzian Approach to Analyzing Mechanical Properties of Endothelial Cells Probed by Atomic Force Microscopy

[+] Author and Article Information
Kevin D. Costa1

Department of Biomedical Engineering,  Columbia University, New York, NYkdc17@columbia.edu

Alan J. Sim

Department of Biomedical Engineering,  Columbia University, New York, NY

Frank C-P. Yin

Department of Biomedical Engineering,  Washington University, St. Louis, MO

1

Corresponding author.

J Biomech Eng 128(2), 176-184 (Nov 18, 2005) (9 pages) doi:10.1115/1.2165690 History: Received September 20, 2004; Revised November 18, 2005

## Abstract

Detailed measurements of cell material properties are required for understanding how cells respond to their mechanical environment. Atomic force microscopy (AFM) is an increasingly popular measurement technique that uniquely combines subcellular mechanical testing with high-resolution imaging. However, the standard method of analyzing AFM indentation data is based on a simplified “Hertz” theory that requires unrealistic assumptions about cell indentation experiments. The objective of this study was to utilize an alternative “pointwise modulus” approach, that relaxes several of these assumptions, to examine subcellular mechanics of cultured human aortic endothelial cells (HAECs). Data from indentations in $2‐to5‐μm$ square regions of cytoplasm reveal at least two mechanically distinct populations of cellular material. Indentations colocalized with prominent linear structures in AFM images exhibited depth-dependent variation of the apparent pointwise elastic modulus that was not observed at adjacent locations devoid of such structures. The average pointwise modulus at an arbitrary indentation depth of $200nm$ was $5.6±3.5kPa$ and $1.5±0.76kPa$ ($mean±SD$, $n=7$) for these two material populations, respectively. The linear structures in AFM images were identified by fluorescence microscopy as bundles of f-actin, or stress fibers. After treatment with $4μM$ cytochalasin B, HAECs behaved like a homogeneous linear elastic material with an apparent modulus of $0.89±0.46kPa$. These findings reveal complex mechanical behavior specifically associated with actin stress fibers that is not accurately described using the standard Hertz analysis, and may impact how HAECs interact with their mechanical environment.

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## Figures

Figure 1

(A) Light micrograph (100×) of isolated HAEC on a tissue culture dish at room temperature. AFM probe is visible in the upper right corner. Squares indicate approximate regions imaged with AFM, and are not drawn to scale. (B) Contact mode AFM deflection image of the 10×10μm region of the living cell indicated by a solid square in panel (A), and (C) of the 25×25μm region of the cell indicated by a broken square in panel (A). (D) and (E) AFM deflection images of the same regions of the cell after fixation for 30min with 3.7% formaldehyde in isotonic saline. Scan rate was 100μm∕s in all images.

Figure 2

(A) Mosaic of seven 60×60μm AFM contact mode height images obtained after fixation of the HAEC in Fig. 1. (B) Fluorescent micrograph of the same cell stained with rhodamine-phalloidin to identify f-actin. There is a clear correlation between the filamentous structures identified by AFM and actin fiber bundles in the cytoskeleton, some of which are distinctly nonlinear (arrows).

Figure 3

Example AFM indentation data obtained from two locations spaced 1.5 microns apart on a living HAEC under untreated conditions. (A) Advance portion of raw force curve with estimated contact point indicated by the vertical bar. (B) Associated indentation force versus depth data. Solid line shows least-squares fit using Hertz model (see text for details), with inset showing the distribution of residuals. (C) Stiffness plot of elastic modulus versus depth obtained using the pointwise analysis (symbols) and from the curve fit in panel (B) (solid line).

Figure 4

Contact mode AFM deflection images (15×15μm) of the peripheral margin of a human aortic endothelial cell on a tissue culture dish (A) before and (B) after 45min treatment with 4μM cytochalasin B. The white square on each image indicates a 3.5×3.5μm region where an 8×8 array of 64 indentations was performed to measure local cell mechanical properties. (C) Pointwise modulus versus indentation depth pre (black) and post (gray) cytochalasin-treatment from the square region identified in panels (A) and (B). All curves are shown on the same scale of 0–20kPa for the modulus and 0–400nm for depth. Note band of depth-dependent modulus values coincident with the large diagonal stress fiber in panel (A). The four stiffness plots with heavy black frames correspond to the four locations identified by small black squares in panels (A) and (B).

Figure 5

Contact mode AFM images (15×15μm) of untreated HAECs (A), (C), (D) and after 45min treatment with 4μM cytochalasin B (B). Superposed on each image is a gray scale map of the pointwise elastic modulus extracted at an indentation depth of 200nm obtained from an array of 64 force curves as described in Fig. 4 with the region size and location drawn to scale. Histograms show the distribution of values from each corresponding modulus map, aligned with the scale bar for the modulus values (0–16kPa). Note alignment of stiff regions with fibrous structures in untreated cells, and a broad or bimodal distribution of modulus values compared to the cytochalasin-treated cell.

Figure 6

Histograms of HAEC pointwise modulus (at D=200nm) in bins of 0.5kPa. (A) Results from untreated cells (64 measurements per cell, n=7 cells) show a bimodal distribution suggesting two populations of material within the cell. The fitted sum of two normal distributions (heavy line) yielded values of 1.5±0.76kPa and 5.6±3.5kPa for the means and standard deviations of the two populations (individual distributions in fine lines). (B) Corresponding results from cells treated with 4μM cytochalasin B (n=3 cells) show a distribution indicative of a single population of cellular material with an average modulus of 0.89±0.46kPa.

Figure 7

Contact mode AFM images (15×15μm) of a living HAEC before (A) and immediately following (B) a 64-indentation force volume array in the region indicated by the white square, demonstrating reasonable stability of cytoskeletal structures during such experiments.

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