Energy injection in an epithelial cell monolayer indicated by negative viscosity

TL;DR

This study studies how energy is transferred by developing a method to measure the effective viscosity from the shear stresses and strain rates within an epithelial cell monolayer and shows that negative effective viscosity is a useful means of quantifying the flow of energy in living matter.

Abstract

Epithelial tissues are driven out of thermodynamic equilibrium by internally generated forces, causing complex patterns of motion. Even when both the forces and motion are measurable, it is not yet possible to relate the two, because the sources of energy injection and dissipation are often unclear. Here, we study how energy is transferred by developing a method to measure the effective viscosity from the shear stresses and strain rates within an epithelial cell monolayer. Interestingly, there emerged multicellular regions in which the relationship between shear stress and shear strain rate was negatively proportional, indicating a negative effective viscosity. The negative effective viscosity occurred in regions wherein cell stresses were less efficient at producing tissue deformations compared to regions of positive effective viscosity. Regions of negative effective viscosity consistently exhibited greater cell speed and vorticity, and the cells had elevated metabolic activity, reflecting an increased energy demand in these cells. Our study shows that negative effective viscosity is a useful means of quantifying the flow of energy in living matter.

Figures (14)

• Figure 1: Viscosity in the cell monolayer. (a) Phase contrast image of a cell monolayer overlaid with colors indicating regions of positive (blue) and negative (yellow) viscosity. (b) Heat map of shear stress, $(\sigma_1 - \sigma_1)/2$. (c) Heat map of shear strain rate $(\dot{\varepsilon}_1 - \dot{\varepsilon}_2)/2$. (d) Spatial correlation of shear strain rate (gray) and shear stress (turquoise). The solid and dashed lines indicate averages of the respective spatial correlations; the dotted line indicates the size of a single cell. (e) Scatter plot of shear stress and shear strain rate within a representative window showing positive viscosity (Pearson's correlation coefficient: 0.55). The slope of the line is $25.9 \pm 6.9$ Pa$\cdot$hr (slope $\pm$ standard error from linear regression). (f) Scatter plot of shear stress and shear strain rate within a representative window showing negative viscosity (Pearson's correlation coefficient: -0.62). The slope of the line is $-33.6 \pm 7.3$ Pa$\cdot$hr (slope $\pm$ standard error from linear regression). (g) Distribution of negative viscosity region size over 6 cell islands over 15 hr of imaging. Inset: Average size of regions of negative viscosity. (h) Percentage of the total island area exhibiting negative viscosity over time. (i) Percentage of the cells exhibiting negative viscosity against distance from the center of the island. (j) Histogram showing apparent viscosity over 6 cell islands over 15 hr of imaging. In panels g--i, gray dots and gray lines are data from different cell monolayers; black lines indicate means.
• Figure 2: Stresses and strain rates in regions of positive and negative viscosity. (a) Histogram of $(\sigma_1-\sigma_2)/2$ in regions of positive and negative viscosity. (b) Histogram of $(\sigma_1+\sigma_2)/2$ in regions of positive and negative viscosity. (c) Histogram of $(\dot{\varepsilon}_1 - \dot{\varepsilon}_2)/2$ in regions of positive and negative viscosity. (d) Histogram of $(\dot{\varepsilon}_1 + \dot{\varepsilon}_2)/2$ in regions of positive and negative viscosity. (e) Depiction of the stresses (black and yellow arrows) applied by a cell (light gray) onto its neighbors (dark gray). Negative viscosity would occur if the orientation of the second principal stress, $\sigma_2$ (yellow arrows), aligns with the orientation of the first principal strain rate, $\dot{\varepsilon}_1$ (green arrows). The difference between orientations of $\sigma_2$ and $\dot{\varepsilon_1}$ is indicated by angle $\theta$. (f) Image of a negative viscosity region overlaid with lines depicting the orientations of $\sigma_2$ (yellow) and $\dot{\varepsilon}_1$ (green). (g) Histogram of angle $\theta$ between orientations of $\sigma_2$ and $\dot{\varepsilon}_1$ in regions of negative and positive viscosity ($p < 0.0001$, Kolmogorov-Smirnov test comparing the two distributions). (h) Depiction of the stresses (black and yellow arrows) applied by a cell (light gray) onto its neighbors (dark gray). The principal strain rates are shown by the solid green arrows. The strain rate tensor was projected into orientation of stress, and the dashed arrows show the diagonal components of the projected strain rates, $\dot{\varepsilon}_{\sigma_1}$ and $\dot{\varepsilon}_{\sigma_2}$. (i) Distribution of $-\dot{\varepsilon}_{\sigma_1}$ in well-defined regions of positive and negative viscosity. (j) Mean of $-\dot{\varepsilon}_{\sigma_1}$ for well-defined regions of positive and negative viscosity ($p < 0.0001$, two-sample t-test). Each dot represents a cell island. (k) Distribution of $\dot{\varepsilon}_{\sigma_2}$ in well-defined regions of positive and negative viscosity. (l) Mean of $\dot{\varepsilon}_{\sigma_2}$ for well-defined regions of positive and negative viscosity ($p < 0.0001$, two-sample t-test). Each dot represents a cell island. All histograms are over 6 cell islands and 15 hr of imaging; error bars represent the standard deviation of the probability values for each bin, calculated across all cell islands.
• Figure 3: Altering stresses and flow rates does not alter the fraction of cells with negative viscosity. (a) Phase contrast image of a cell monolayer (left) overlaid with regions of positive (blue) and negative (yellow) viscosity. Heat maps of maximum shear stress (center) and maximum shear strain rate (right). Data are shown for control cells (top row) and cells treated with CN03 (middle row) and cytochalasin D (cytoD, bottom row). (b) Shear stress for treatments and their respective vehicle controls. (CN03 $p = 0.012$, cytoD $p = 0.039$, two-sample t-tests). As described in methods, the control for cytoD used a higher concentration of fetal bovine serum (FBS), 10%. (c) Shear strain rate for treatments and their respective vehicle controls (CN03 $p = 0.002$, cytoD $p < 0.0001$, two-sample t-tests). (d) Percent negative viscosity within the monolayer for control and treatment cases (CN03 $p = 0.5$, cytoD $p = 0.99$, two sample t-tests). (e) Ratio of RMS velocity in regions of negative viscosity to positive viscosity for control and treatment cases (control $p < 0.0001$, CN03 $p < 0.0001$, control 10$\%$ FBS $p = 0.002$, cytoD $p < 0.0001$, one-sample t-tests in comparison to 1). (f) Ratio of RMS vorticity, $\omega$, in regions of negative viscosity to positive viscosity for control and treatment cases (control $p < 0.0001$, CN03 $p < 0.0001$, control 10$\%$ FBS $p = 0.02$, cytoD $p < 0.0001$, one-sample t-tests in comparison to 1). (g) Characteristic size of a vortex, given by RMS v / RMS $\omega$, for the control and treatment cases (CN03 $p = 0.11$, cytoD $p = 0.09$, two sample t-tests). For all panels, a dot indicates the mean of a cell island over time, and black bars indicate means.
• Figure 4: Regions of negative viscosity are more metabolically active. (a) Fluorescent image of cell island stained with 2-NBDG (top) and overlaid with map depicting positive (blue) and negative (yellow) viscosity (bottom). The black box is an enlarged view, showing the outlines of individual cells. (b) Fluorescent image of cell island stained with TMRE (top) and overlaid with map depicting positive (blue) and negative (yellow) viscosity (bottom). The black box is an enlarged view, showing the outlines of individual cells. (c) Ratio of 2-NBDG and TMRE fluorescent intensity in regions of negative viscosity to regions of positive viscosity (2-NBDG $p < 0.0001$, TMRE $p < 0.0001$, one-sample t-tests in comparison to 1). (d) Fluorescent intensity of 2-NBDG and TMRE was normalized such that the average was unity for each cell island and then plotted against distance from the center. The lines show data averaged over 12 islands treated with 2-NBDG and 11 islands treated with TMRE. (e) Spatial correlation of TMRE and 2-NBDG fluorescent intensity. The dotted line indicates the size of a single cell. (f) Normalized fraction of area having negative viscosity before and after metabolic inhibition (MI). All values were normalized by the mean of times $\in (-50,0)$ min. The metabolic inhibition occurred at $t=0$ min. Lines represent the mean and error bars represent the standard deviation. (g) Fractional change in area of negative viscosity from initial time point to last time point (Control $p = 0.396$, MI $p = 0.004$, one-sample t-test in comparison to 1). (h) Distribution of viscosity lifetime in regions of positive and negative viscosity. Inset: Average lifetime of regions of positive and negative viscosity. Each dot represents a different cell island. (i) Fluorescent intensity of TMRE over time. Each line indicates a randomly chosen $62 \times 62$ µm$^2$ region of interest (ROI). Gray and blue line segments indicate times over which the ROI had positive and negative viscosity, respectively. Fluorescent intensity in each ROI was normalized by the mean over all time points in that ROI. The sign of viscosity was determined based on the sign of the majority of grid points within the ROI at each time point. (j) Average slope of normalized fluorescent intensity for positive and negative viscosity line segments of each ROI ($p < 0.0001$, paired sample t-test). Dots in panels c, f, and g indicate averages over a cell island. Black bars indicate means.
• Figure 5: Shear stress and shear strain rate across the entire island is uncorrelated. The representative scatter plot shows $(\sigma_{1}-\sigma_{2})/2$ against $(\dot{\varepsilon}_1-\dot{\varepsilon}_2)/2$ across an entire cell island at one point in time.