Comparison of viscoelastic properties of cancer and normal thyroid cells on different stiffness substrates

Abstract

We used atomic force microscopy (AFM) technique to measure the viscoelastic response of cancer and normal thyroid cells on different stiffness polyacrylamide gels. After applying a step in contact we recorded the stress relaxation of cells in order to measure their viscous and elastic properties. With the help of an extended version of the Hertz model, we could quantify for the first time by AFM the elastic modulus and the dynamic viscosity of cells on substrates with different stiffnesses. We have cultured anaplastic carcinoma and normal thyroid cells on three different substrates: polyacrylamide gels with elastic modulus in a range of 3–5 and 30–40 kPa and “infinitely” stiff Petri dishes. Whereas normal thyroid cells adapted their mechanical properties to different stiffness substrates, cancer cells were less affected by the surrounding stiffness. Normal cells changed the elastic modulus from 1.2 to 1.6 and to 2.6 kPa with increasing substrate stiffness; the dynamic viscosity values varied from 230 to 515 and to 470 Pa·s, accordingly. By contrast, the values for cancer cells were rather constant regardless of substrate stiffness (in average the elastic modulus was 1.3 kPa and the dynamic viscosity was 300 Pa·s). This difference in sensing and reacting to the mechanical properties of the substrate shows that normal and cancer cells interact differently with the neighboring tissue, which may be related to the ability of cancer cells to form metastases.

Appendix

Cell actin staining

Forty-eight hours after seeding on gels and Petri dishes, cells were fixed with 3.7 % formaldehyde for 15 min and then permeabilized with 0.1 % Triton X100 for 15 min. Samples were washed with PBS after each step and then incubated with a rhodamine phalloidin solution (5:200 dilution in PBS) for 30 min at 20 °C. Finally, cells were stored in PBS at 4 °C until image acquisition. An Axiovert 135 TV epifluorescence microscope (Carl Zeiss MicroImaging GmbH, Germany) was used to observe cells and collect fluorescent images (Fig. 5).

Fig. 5

Fluorescent images of normal and cancer thyroidal cells labeled with rhodamine phalloidin (actin cytoskeleton). The upper panels show normal cells seeded on soft (a) and stiff (b) PA gels and on Petri dish (c). The lower panels show cancer cells, seeded on soft (d) and stiff (e) PA gels and on Petri dish (f). Scale bars are 50 μm

Analysis of stress relaxation data

To analyze stress relaxation data, we assume that the sample can be described with the general linear solid model (Fig. 6).

Fig. 6

Standard linear solid model. The sample is modeled by a Zener element, where a spring k ₁ is in parallel to a Maxwell element, consisting of a spring k ₂ and a viscous damping element f. The cantilever is characterized by its spring constant k _c . Viscous (hydrodynamic) damping of the cantilever is neglected here, due to the slow stress relaxation of the sample, which is the predominant contribution

The force balance is given in the static case (directly after the step) by the following equation:

$$\begin{aligned} F_{c} =\, & F_{1} + F_{2} \\ F_{2} =\, & k_{c} *d - k_{1} *\delta \\ \end{aligned}$$

(1)

where k _c is the force exerted by the cantilever, k₁ and k₂ is the forces exerted by the two springs. After relaxation F₂ will be zero.

The force in the Maxwell element has to obey the following dynamic equation:

$$\dot{\delta } = \frac{{\dot{F}_{2} }}{{k_{2} }} - \frac{{F_{2} }}{f}$$

(2)

By fitting an exponential function (that was the more adequate to describe our results, see Fig. 7) to the deflection data, we determined the fit parameters ∆d, a, and τ.

Fig. 7

Stress relaxation after applying a loading step in z height at t = 1.5 s and an unloading step at t = 2.0 s. The indentation is calculated as the difference between z height and deflection. The deflection data are fitted with an exponential function, which will give k ₁, k ₂ and f as results

In details, before applying the step the force equilibrium will be:

$$k_{c} *d_{1} = k_{s} *\delta_{1} = k_{s} *\left( {z_{1} - d_{1} } \right)$$

(3)

where z₁, d₁ and δ ₁ are the z-position, the deflection and the indentation in contact with the sample, respectively. Since the forces are in equilibrium at this point, we can simplify our calculations such that:

$$d_{1} = z_{1} = \delta_{1} = 0$$

(4)

When applying a z-step, the z-height will change to:

$$z_{2} = z_{1} + \Delta z = \Delta z$$

(5)

After relaxation, the deflection d ₂ will be:

$$d_{2} = d_{1} + \Delta d = \Delta d$$

(6)

The spring k ₂ will be relaxed due to the creep of the viscous damping element f, so the force balance looks like:

$$\begin{aligned} k_{c} *d_{2} =\, & k_{1} *\delta_{2} = k_{1} *\left( {z_{2} - d_{2} } \right) \\ k_{c} *\Delta d =\, & k_{1} \left( {\Delta z - \Delta d} \right) \\ \end{aligned}$$

(7)

So, we can derive the spring constant k₁ from the measurable quantities ∆d and ∆z:

$$k_{1} = k_{c} \frac{\Delta d}{\Delta z - \Delta d}$$

(8)

After the step the force balance will be:

$$\begin{aligned} k_{c} *d_{3} = & \left( {k_{1} + k_{2} } \right)*\delta_{3} \\ k_{c} \left( {\Delta d + a} \right) = & \left( {k_{1} + k_{2} } \right)*\left( {z_{2} - d_{3} } \right) \\ k_{c} \left( {\Delta d + a} \right) = & \left( {k_{1} + k_{2} } \right)*\left( {\Delta z - \Delta d - a} \right) \\ \end{aligned}$$

(9)

Therefore we can derive the spring constant k₂

$$k_{2} = k_{c} \frac{\Delta d + a}{\Delta z - \Delta d - a} - k_{1}$$

(10)

To describe the stress relaxation, we use the following relaxation process ansatz:

$$d = d_{2} - d_{1} + a*e^{{ - {\raise0.7ex\hbox{$t$} \!\mathord{\left/ {\vphantom {t \tau }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\tau $}}}} = \Delta d + a*e^{{ - {\raise0.7ex\hbox{$t$} \!\mathord{\left/ {\vphantom {t \tau }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\tau $}}}}$$

(11)

Therefore, Eq. (2) will change to:

$$\dot{\delta } = a*\frac{1}{\tau }*e^{{ - {\raise0.7ex\hbox{$t$} \!\mathord{\left/ {\vphantom {t \tau }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\tau $}}}}$$

(12)

From Eqs. (1), (11) and (12) we can get:

$$f = k_{2} *\tau *\frac{{k_{c} + k_{1} }}{{k_{2} + k_{c} + k_{1} }}$$

(13)

The time constant τ is not the true intrinsic relaxation time τ* of the Zener element, but the apparent relaxation time constant of the Zener element plus the cantilever spring. The intrinsic time constant τ* is given by:

$$\tau ^{*} = \frac{f}{{k_{2} }} = \tau *\frac{{k_{c} + k_{1} }}{{k_{2} + k_{c} + k_{1} }}$$

(14)

Calculating elastic moduli and dynamic viscosity

When using pyramidal indenters, the elastic response is typically described by the Hertz model, which basically considers the increasing contact area when indentation increases.

In the case of a pyramidal tip with opening angle α, elastic modulus E and Poisson ratio ν, we find the following relation between indentation δ and loading force F:

$$F = \frac{1}{\sqrt 2 }*\frac{E}{{1 - \nu^{2} }}*\tan \alpha *\delta^{2}$$

(15)

The spring constant of a sample showing a Hertzian response can be found by taking the derivative. This force constant will depend on force F₀ or indentation δ ₀ at which it is calculated (or measured):

$$k_{Hertz} = \left. {\frac{\partial F}{\partial \delta }} \right|_{{\delta = \delta_{0} }} = 2*\frac{1}{\sqrt 2 }*\frac{E}{{1 - \nu^{2} }}*\tan \alpha *\delta_{0}$$

(16)

Since the indentation is not measured directly, but the deflection is, we use Eq. 15 to replace indentation by force, which is deflection times the cantilever’s force constant:

$$\delta_{0} = \sqrt {F_{0} *\sqrt 2 *\frac{{1 - v^{2} }}{E}*\frac{1}{\tan \alpha }} = \sqrt {k_{c} *d_{0} *\sqrt 2 *\frac{{1 - v^{2} }}{E}*\frac{1}{\tan \alpha }}$$

(17)

So, we can replace δ ₀ in Eq. 16 and get the force constant as a function of the measurable quantity d ₀, the deflection value at which the force constant is measured or calculated:

$$\begin{aligned} k_{Hertz} = & 2*\frac{1}{\sqrt 2 }*\frac{E}{{1 - v^{2} }}*\tan \alpha *\sqrt {F_{0} *\sqrt 2 *\frac{{1 - v^{2} }}{E}*\frac{1}{\tan \alpha }} \\ k_{Hertz} = & \sqrt {2*\sqrt 2 *\frac{E}{{1 - v^{2} }}*\tan \alpha *F_{0} } \\ \end{aligned}$$

(18)

In a stress relaxation experiment, we actually measure the spring constants k₁ and k₂ of our sample, so we can invert Eq. 18 to convert spring constants in the corresponding moduli:

$$E_{ 1/2} = k_{1/2}^{2} * \frac{1}{2*\sqrt 2 }* \frac{1}{\tan \alpha }* \left( {1 - v^{2} } \right) * \frac{1}{{F_{0} }}$$

(19)

The dynamic viscosity η in the linear standard solid model can be written as:

$$\eta = E_{2} * \tau^{*}$$

(20)

where $\tau^{*}$ is the intrinsic relaxation time as defined in Eq. 14.

Numerical results of cell moduli

Tables 1 and 2 show the numerical results presented in Fig. 3 in the main text for normal and cancer cells, respectively. Data are presented as median values and error bars show the 25th and 75th percentiles.

Table 1 Numerical results of moduli of normal thyroidal cells, presented in Fig. 3 of the main text (with blue markers): Young modulus approach (a.) and retract (r.) calculated from force curves; elastic modulus and dynamic viscosity calculated from the loading (l.) and unloading (u.) steps in stress relaxation experiments

Table 2 Numerical results of moduli of cancer thyroidal cells, presented in Fig. 3 of the main text (with red markers): Young’s modulus approach (a.) and retract (r.) calculated from force curves; elastic modulus and dynamic viscosity calculated from the loading (l.) and unloading (u.) steps in stress relaxation experiments

Statistical analysis of moduli between substrates

Differences between the moduli presented in Fig. 3 were determined with the Wilcoxon signed-rank test, calculated in IGOR, and considered significant when p < 0.05. We first calculated the median values of each force map, grouped them by cell type (normal and cancer) and substrate (soft and stiff gels, Petri dish), and checked whether the differences between groups were significant. Since the distribution of the data is not Gaussian, we could not refer to a t test, but had to use a Wilcoxon rank test, which has no assumptions on the distribution of data. We present here only significance values when comparing data from the same cell type on different substrates, since the hypothesis is that cancer and normal cells react differently on alterations of substrate stiffness. On a particular substrate, there may be (and actually are) differences also between the cell types, which we ignore here in the presentation. Significance data are presented as 2D matrixes, where each colored box presents one comparison (soft vs. stiff; soft vs. Petri, stiff vs. Petri). Since this matrix will be symmetrical, we plotted in the lower left corner the values for normal cells and in the upper right corner those for cancer cells. The significance values are color-coded such that all values below 0.05 show up as green, and all others as black. In this way, we can easily pick up differences between cancer and normal cells. Six matrixes are reported in Fig. 8 for each quantity presented in Fig. 3 (Young’s modulus from approach and retract, elastic modulus, and dynamic viscosity, each from loading and unloading steps).

Fig. 8

Statistical analysis on Young’s moduli approach (a) and retract (b) from conventional force curves; elastic moduli E, loading (c) and unloading (d) and dynamic viscosity η, loading (e) and unloading (f) from stress relaxation experiments. In each matrix, cancer and normal cell cases are enclosed by red and blue lines, in the upper right and lower left corners, respectively. Differences are considered significant when p < 0.05 (green boxes) and not significant when p > 0.05 (black boxes)

It is apparent that for all quantities we have calculated, the values from normal cells are significantly different, virtually between all substrate combinations, whereas in the case of cancer cells the opposite behavior is true: virtually all quantities show non-significant differences between substrates.