If cell mechanics can be described by elastic modulus: study of different models and probes used in indentation experiments

Abstract

Here we investigated the question whether cells, being highly heterogeneous objects, could be described with the elastic modulus (effective Young's modulus) in a self-consistent way. We performed a comparative analysis of the elastic modulus derived from the indentation data obtained with atomic force microscopy (AFM) on human cervical epithelial cells (both normal and cancerous). Both sharp (cone) and dull (2500-nm radius sphere) AFM probes were used. The indentation data were processed through different elastic models. The cell was approximated as a homogeneous elastic medium that had either 1), smooth hemispherical boundary (Hertz/Sneddon models) or 2), the boundary covered with a layer of glycocalyx and membrane protrusions ("brush" models). Consistency of these approximations was investigated. Specifically, we tested the independence of the elastic modulus of the indentation depth, which is assumed in these models. We demonstrated that only one model showed consistency in treating cells as a homogeneous elastic medium, namely, the brush model, when processing the indentation data collected with the dull AFM probe. The elastic modulus demonstrated strong depth dependence in all models: Hertz/Sneddon models (no brush taken into account), and when the brush model was applied to the data collected with sharp conical probes. We conclude that it is possible to describe the elastic properties of the cell body by means of an effective elastic modulus, used in a self-consistent way, when using the brush model to analyze data collected with a dull AFM probe. The nature of these results is discussed.

Figure 1

A schematic of AFM probe-cell surface interaction. Brush layer is shown. The value Z is the relative piezo position of the cantilever, d is the cantilever deflection, Z₀ is the nondeformed position of the sample surface, i is the deformation of the sample, and Z = 0 is for the maximum deflection assigned by the AFM user. The value h is the separation between sample and the AFM probe.

Figure 2

An example of processing raw data, deflection of the AFM cantilever versus vertical position of the AFM scanner (d versus Z) collected with a spherical AFM indenter (approaching curve). (a) The regions of the curve fitted with the Hertz and brush model are shown. The fitting region in the Hertz model starts from the contact, whereas the region for the brush model is near the maximum load where brush is squeezed. (b) Fitting the brush force curve with the steric brush model (solid line); see Eq. 6. To see this figure in color, go online.

Figure 3

Brush models, with representative examples of the dependence of the elastic modulus (shown in kPa) on the indentation depth (shown in nanometers) of human cervical epithelial cells when using (a) conical and (b) spherical AFM indenters. To see this figure in color, go online.

Figure 4

The elastic modulus derived in the Sneddon/Hertz models. Representative examples of the dependence of the elastic modulus (shown in kPa) on the indentation depth (shown in nanometers) of human cervical epithelial cells when using conical (a, left) and spherical (b, right) AFM probes (note: brush not taken into account). To see this figure in color, go online.

Figure 5

Comparison of the elastic modulus on the load force derived in (a) the Hertz model and (b) the brush model (the same raw data was used for the both models). To see this figure in color, go online.

Figure 6

The case of the spherical indenter. (a–f) Histograms of distributions of the elastic modulus of normal and cancer cells derived for various indentations in different models. (c and f) Histograms showing behavior of an ideal elastic material under the same conditions. To see this figure in color, go online.

Figure 7

The case of the conical indenter. (a–f) Histograms of distributions of the elastic modulus of normal and cancer cells derived for different indentations in different models. (c and f) Histograms showing behavior of ideal elastic materials under the same conditions. To see this figure in color, go online.

Figure 8

The schematics of the histograms of Figs. 6 and 7 showing the actual depth dependence for the modulus measurement at a single surface point. Examples of cancer cells are shown: (a) for the case of Fig. 7f (an ideal elastic material, cone indenter, Sneddon model); (b) for the case of Fig. 7d (cone indenter, Sneddon model); (c) for the case of Fig. 6d (spherical indenter, Hertz model); and (d) for the case of Fig. 6e (spherical indenter, brush model). To see this figure in color, go online.