Quantifying cell-to-cell variation in power-law rheology

Abstract

Among individual cells of the same source and type, the complex shear modulus G(∗) exhibits a large log-normal distribution that is the result of spatial, temporal, and intrinsic variations. Such large distributions complicate the statistical evaluation of pharmacological treatments and the comparison of different cell states. However, little is known about the characteristic features of cell-to-cell variation. In this study, we investigated how this variation depends on the spatial location within the cell and on the actin filament cytoskeleton, the organization of which strongly influences cell mechanics. By mechanically probing fibroblasts arranged on a microarray, via atomic force microscopy, we observed that the standard deviation σ of G(∗) was significantly reduced among cells in which actin filaments were depolymerized. The parameter σ also exhibited a subcellular spatial dependence. Based on our findings regarding the frequency dependence of σ of the storage modulus G('), we proposed two types of cell-to-cell variation in G(') that arise from the purely elastic and the frequency-dependent components in terms of the soft glassy rheology model of cell deformability. We concluded that the latter inherent cell-to-cell variation can be reduced greatly by disrupting actin networks, by probing at locations within the cell nucleus boundaries distant from the cell center, and by measuring at high loading frequencies.

Figure 1

(a) Fluorescence image of nuclei of NIH3T3 cells deposited in microarray wells and cultured for 12 h using confocal microscopy. The cell nuclei were stained with DAPI. (b) Schematic of the AFM force modulation with a microarray substrate, on which living cells were arranged and cultured. (c) Measurements of the effect of cytoD on the $G^{\ast }$ of cells. The untreated cells were measured at the center of wells and the same cells treated with cytoD were measured again at the same location. (d) Measurements of the spatial dependence of $G^{\ast }$ of cells at different locations of the center and away from the center of wells.

Figure 2

Distributions of the storage $G^{\prime }$ (left) and loss $G^{″}$ (right) moduli of untreated cells (white, n = 87) and cytoD-treated cells (gray, n = 87) in microarray wells at different frequencies: (a) 2, (b) 50, and (c) 200 Hz. The solid and dashed lines represent the fitted results of untreated and treated cells, respectively, using a log-normal distribution function.

Figure 3

Frequency dependences of ${\overline{G}}^{\ast }$(${\overline{G}}^{\prime }$ (a) and ${\overline{G}}^{″}$ (b)) of untreated (circle) and treated (square) cells. Solid lines in (a) and (b) represent the fitted results to Eq. 3. The mean squared error is 25.8 (a). The point where the curves of ${\overline{G}}^{\prime }$ intersect is defined as ${\overline{G}}^{\prime }={\overline{g}}_0$ at $f={\overline{\Phi }}_0$. Frequency dependency of ${\sigma }_{\ln G^{\prime }}$ (c) and ${\sigma }_{\ln G^{″}}$ (d) of untreated (circle) and treated (square) cells. Solid lines in (c) and (d) represent the fitted results using Eqs. 7 and S19, respectively (see Discussion). The mean squared error is 0.011 (c).

Figure 4

Distributions of (a) G₀ on a logarithmic scale, (b) α on a linear scale, and (c) μ on a logarithmic scale of untreated (white) and treated (gray) cells. Inset in (c) shows the distribution of μ on a linear scale. Solid and dashed lines represent the fitted results of untreated and treated cells, respectively, using a log-normal distribution function (a and c) and to a normal distribution function (b and inset in c).

Figure 5

Distributions of G′ (left) and G″(right) moduli of cells measured at center (white, n = 160) and off-center (gray, n = 160) locations of microarray wells at different frequencies: (a) 2, (b) 50, and (c) 200 Hz. Solid and dashed lines represent fitted results of cells measured at center and off-center locations, respectively, using a log-normal distribution function.

Figure 6

Frequency dependences of ${\overline{G}}^{\ast }$(${\overline{G}}^{\prime }$ (a) and ${\overline{G}}^{″}$ (b)) of cells measured at center (circle) and off-center (triangle) locations of wells. Solid lines in (a) and (b) represent the fitted results to Eq. 3. The mean squared error is 13.0 (a). The point where the curves of ${\overline{G}}^{\prime }$ intersect is defined as ${\overline{G}}^{\prime }={\overline{g}}_0$ at $f={\overline{\Phi }}_0$. Frequency dependences of ${\sigma }_{\ln G^{\prime }}$, (c) and ${\sigma }_{\ln G^{″}}$, (d) of cells measured at center (circle) and off-center (triangle) locations of wells. Solid lines in (c) and (d) represent the fitted results using Eqs. 7 and S19, respectively (see Discussion). The mean squared error is 0.014 (c).

Figure 7

Distributions of (a) lnG₀, (b) α and (c) lnμ of cells measured at center (white) and off-center (gray) locations of wells. Solid and dashed lines represent the fitted results of cells measured at center and off-center locations, respectively, using a log-normal distribution function (a and c) and a normal distribution function (b).

Figure 8

$G^{\prime }$ measured at the center of wells, $G_{\mathrm{center}}^{\prime }$, vs. $G^{\prime }$ measured away from the center of wells, $G_{\mathrm{off}\text{-}\mathrm{center}}^{\prime }$ (left) and $G^{″}$ measured at the center of wells, $G_{\mathrm{center}}^{″}$, vs. $G^{″}$ measured away from the center of wells, $G_{\mathrm{off}\text{-}\mathrm{center}}^{″}$ (right), at different frequencies: (a) 5, (b) 50, and (c) 200 Hz. The data are the same as those shown in Figs. 5–7.

Figure 9

Plots of lnG₀ vs. α (a) and lnG₀ vs. lnμ (b) of cells measured at center (open circle) and off-center (solid circle) locations of cells, which are shown in Figs. 5–7. The solid lines in (a) and (b) represents the fitted results using Eq. 6 and a linear function, respectively.

Figure 10

${\tilde{\sigma }}_{\ln G^{\prime }}$, which represents ${\sigma }_{\ln G^{\prime }}-{\sigma }_{\ln g_0}$ as a function of lnf. The results obtained from two cell samples shown in Fig. 3c and in Fig. 6c are replotted: One sample is untreated (solid rectangle) and treated (open rectangle) cells measured at the center of wells, whereas the other is untreated cells measured at the center (solid triangle) and away from the center (open triangle) of wells. Solid lines represent the fitted results using Eq. 7.

Figure 11

Schematic of $G^{\prime }$ of untreated cells (a) and cytoD-treated cells (b) at different frequencies. The cell-to-cell variation of $G^{\prime }$ varies depending on intracellular locations: the distribution narrows when changing from cell center to cell nucleus boundaries. The spatial component of cell-to-cell variation of $G^{\prime }$ between the untreated and treated cells decreases with increasing f, and consequently both cells become spatially homogeneous at $f={\overline{\Phi }}_0$ beyond the SGR region (see Eq. 7), but the cell-to-cell variation still exists at $f={\overline{\Phi }}_0$. The spatial variation of $G^{\prime }$ for the untreated cells in the SGR region is larger than that for treated cells. One experimental condition is that $G_{\mathrm{a}}^{\prime }$ (σ_II) and $G_{\mathrm{b}}^{\prime }$ (σ_I) represent the values measured at off-center and center locations, respectively, whereas the other $G_{\mathrm{a}}^{\prime }$ (σ_I) and $G_{\mathrm{b}}^{\prime }$ (σ_II) are those of the untreated and treated cells, respectively (c).