Measuring nanoscale viscoelastic parameters of cells directly from AFM force-displacement curves

Abstract

Force-displacement (F-ZIn recent years interest has increased in the measurement of the viscoelastic properties of soft biological samples motivated by their correlation with disease, differentiation, or cellular transformation [1][2][3][4] . A large variety of methods have been introduced for measuring cellular mechanical properties including micropipette aspiration 5,6 , stretching or compression between two microplates [7][8][9] , optical tweezers 10,11 , and magnetic twisting cytometry 12,13 . However, indentation with the atomic force microscope (AFM) remains one of the most popular methods for probing the nanoscale properties of soft samples like cells, tissues and hydrogels [14][15][16][17] . In the AFM, the elastic properties of live cells are usually evaluated from force versus displacement (F-Z) curves. Then the Hertz model or its modifications are applied to the approach part of the F-Z curve to extract Young's modulus (E Hertz ), the elastic parameter used for characterization of the sample's mechanical properties 18,19 . However, such models assume a purely elastic nature of the sample, while in reality most biological samples are viscoelastic. Viscoelasticity is revealed in a clear hysteresis between the approach and retraction parts of curves 20 ; the indentation speed dependence of E Hertz values extracted from force curves with the Hertz model 11 ; the observations of force relaxation at constant indentation depth and the creep at constant loading force 21 . Approaches other than the standard F-Z curves are usually used to obtain the viscoelastic properties of samples with AFM in both the time [22][23][24] and frequency domains [25][26][27][28][29][30] . These generally require modifications in the AFM apparatus and/or in the data acquisition protocol. Equally importantly, each approach has its own sets of measurement uncertainties.If a standard F-Z curve could also be used to quantify viscoelastic properties, it would allow one standard method with well quantified uncertainties 31 to be used for both viscoelasticity and elasticity measurements. This has not been possible to date, we believe, due to the lack of a mathematical/computational framework that allows the post-processing of force-displacement data to extract the relevant viscoelastic constitutive parameters.

Viscoelasticity in simple indentation-cycle experiments: a computational study

“…More advanced approaches involve algorithms for the fitting of the force curves to the preselected viscoelastic models [5,15,16] or algorithms for the direct reconstruction of the relaxation function. In both cases, however, it is useful to obtain experimental curves in a wide range of indentation rates, and thus, it is useful to know how the curves should look for different viscoelastic models.…”

“…1A): cylinder, sphere, and cone (or pyramid, the difference will be only in the geometrical factor). For the indentation displacement-controlled experiment, the Ting's solution could be presented in the following form [5]:…”

“…The MATLAB code based on the previous works [5,29] was used here to obtain a numerical solution of the Ting's model. It calculates the force versus time and the force versus displacement dependencies via numerical differentiation and integration steps, both for the approach and retraction parts of a force curve, the latter involves the numerical calculation of the 1 () tt function by an iterative procedure.…”

“…Most of them are using modifications of the testing protocol by including constant stress or constant strain phases and oscillatory indentation. However, there are also methods to extract viscoelastic properties directly from the simplest indentation experiment presenting a cycle of approach (deepening into the sample) and retraction of the indenter [5,[15][16][17]. Indentation of a viscoelastic body presents a complex problem with time-varying boundary conditions.…”

The role of vimentin in directional migration of rat fibroblasts

“…Using a microsphere instead of a sharp needle allows to better control the geometry of the probe. Young's moduli were calculated using the Image Analysis program (NT‐MDT), and the OriginPro program (Originlab Corp., Northampton, MA) using an in‐house developed script (Efremov, Wang, Hardy, Geahlen, & Raman, ). If the variations of the moduli were more than 100% or a measurement series was captured close to the cell edge, and Young's modulus increased abruptly when the spherical probe touched the substrate rather than the cell (Figure S1c), then all data series (15 force curves) were discarded.…”