Nanorheology of living cells measured by AFM-based force–distance curves

Abstract

Mechanobiology aims to establish functional relationships between the mechanical state of a living a cell and its physiology. The acquisition of force-distance curves with an AFM is by far the dominant method to characterize the nanomechanical properties of living cells. However, theoretical simulations have shown that the contact mechanics models used to determine the Young's modulus from a force-distance curve could be off by a factor 5 from its expected value. The semi-quantitative character arises from the lack of a theory that integrates the AFM data, a realistic viscoelastic model of a cell and its finitethickness. Here, we develop a method to determine the mechanical response of a cell from a force-distance curve. The method incorporates bottom-effect corrections, a power-law rheology model and the deformation history of the cell. It transforms the experimental data into viscoelastic parameters of the cell as a function of the indentation frequency. The quantitative agreement obtained between the experiments performed on living fibroblast cells and the analytical theory supports the use of force-distance curves to measure the nanorheological properties of cells. † Electronic supplementary information (ESI) available: Complete derivation of the analytical expressions for PLR and KV models, coefficients for a conical tip, correspondence between velocities and equivalent indentation frequencies, statistical analysis of the experiments performed on NIH 3T3 fibroblasts. See

Nanomechanical mapping of soft materials with the atomic force microscope: methods, theory and applications

“…It takes values between w = 0 (elastic material) and w = 1 (Newtonian liquid). For a conical tip of half-angle y and a tip's displacement modulated by a waveform, it is deduced 52 FðtÞ ¼ 8 tan y 3p…”

“…Viscoelasticity is an intrinsic feature of the mechanical response of a cell, 51,84,88,89 therefore AFMbased measurements should always be described in terms of viscoelastic models. 49,52,[90][91][92][93][94][95][96][97] A single cell is a heterogeneous, anisotropic and finitethickness material ( Fig. 5a and b).…”

“…84 In addition, the theory to explain how the cell's mechanical properties influence the cantilever's deflection and the tip's indentation is just emerging. 52,93,96 Modeling the mechanical response of a cell is greatly simplified by considering that cells are incompressible (n = 0.5) and are locally isotropic. The first assumption is supported by the mass densities of HeLa cells and fibroblasts which are, respectively, 1.08 and 1.14 g cm À3 .…”

“…Analytical bottom-effect viscoelastic corrections have been deduced for some viscoelastic models. 52 The theoretical approach combines bottom-effect elastic corrections, 44 the correspondence principle between elastic and viscoelastic deformations, 48 and Ting's method to determine the force as a function of the deformation history. 102 In particular, for a power-law rheology model, 52 the force for the tip approaching towards the cell is given by (Fig.…”

“…The combination of force-distance curves obtained with triangular waveforms and Kelvin-Voigt or power-law rheology models enables analytical theories. Those theories were applied to transform the FDCs into viscoelastic parameters 52,95 and FV maps. 66 A short description of those approaches is provided in Section 4.2.…”