Size Matters: Rethinking Hertz Model Interpretation for Cell Mechanics Using AFM

Abstract

Cell mechanics are a biophysical indicator of cell state, such as cancer metastasis, leukocyte activation, and cell cycle progression. Atomic force microscopy (AFM) is a widely used technique to measure cell mechanics, where the Young modulus of a cell is usually derived from the Hertz contact model. However, the Hertz model assumes that the cell is an elastic, isotropic, and homogeneous material and that the indentation is small compared to the cell size. These assumptions neglect the effects of the cytoskeleton, cell size and shape, and cell environment on cell deformation. In this study, we investigated the influence of cell size on the estimated Young's modulus using liposomes as cell models. Liposomes were prepared with different sizes and filled with phosphate buffered saline (PBS) or hyaluronic acid (HA) to mimic the cytoplasm. AFM was used to obtain the force indentation curves and fit them to the Hertz model. We found that the larger the liposome, the lower the estimated Young's modulus for both PBS-filled and HA-filled liposomes. This suggests that the Young modulus obtained from the Hertz model is not only a property of the cell material but also depends on the cell dimensions. Therefore, when comparing or interpreting cell mechanics using the Hertz model, it is essential to account for cell size.

Keywords: Hertz contact model; atomic force microscopy (AFM); cell mechanics; cell stiffness.

Figure A1

Geometry of contact between spherical indenter and cell.

Figure A2

Boussinesq’s problem.

Figure A3

Schematic view of a rigid probe in contact with an elastic half-space.

Figure 1

Overview of study: AFM measurement of liposomes of various sizes: A small diameter, B large diameter; curves fitting with Hertz contact model; estimation of Young’s modulus of elasticity as fitting parameter.

Figure 2

Measured indentation curve for DPPC liposomes in PBS filled with (DPPC) PBS and (DPPC + HA) HA solution. The fit of indentation by the Hertz contact model for the hemispherical AFM tip is shown. Refer to Figure 3 for details on individual liposomes properties (capital letters).

Figure 3

Linear regression plot with 95% confidence intervals (shaded areas) showing measured dependence between the size of DPPC liposomes and Young’s modulus estimated from Hertz model. Measured data along with the range of measured values are shown for liposomes filled with PBS and HA solution, denoted as DPPC and DPPC + HA, respectively. Indentation curves for selected liposomes A–D are shown in Figure 2.