Prestress and Area Compressibility of Actin Cortices Determine the Viscoelastic Response of Living Cells

Shape, dynamics, and viscoelastic properties of eukaryotic cells are primarily governed by a thin, reversibly cross-linked actomyosin cortex located directly beneath the plasma membrane. We obtain time-dependent rheological responses of fibroblasts and MDCK II cells from deformation-relaxation curves using an atomic force microscope to access the dependence of cortex fluidity on prestress. We introduce a viscoelastic model that treats the cell as a composite shell and assumes that relaxation of the cortex follows a power law giving access to cortical prestress, area-compressibility modulus, and the power law exponent (fluidity). Cortex fluidity is modulated by interfering with myosin activity. We find that the power law exponent of the cell cortex decreases with increasing intrinsic prestress and area-compressibility modulus, in accordance with previous finding for isolated actin networks subject to external stress. Extrapolation to zero tension returns the theoretically predicted power law exponent for transiently cross-linked polymer networks. In contrast to the widely used Hertzian mechanics, our model provides viscoelastic parameters independent of indenter geometry and compression velocity.

Figure 1

(a) CLSM image ($xz$ plane) of a MDCK II cell (plasma membrane stained with CellMask) clamped between cantilever and substrate. White lines show the parametrization. (b) Left side: STED image ($xy$ plane) of the cellular cortex (green: actin) of a MDCK II cell. Right side: reconstruction of the cortex [21]. (c) SEM image of a MDCK II cell revealing its cortex structure (right). (d) Mesh size analysis of cortex extracts [21]. (e) Typical compression (i) at $v_0=0.5\mu \mathrm{m}/\mathrm{s}$ followed by force relaxation (ii) of a MDCK II cell and fits according to the Hertz model (blue line, $E_0=450\mathrm{Pa}$, $\beta =0.22$) and the Evans model (red line, $T_0=0.75\mathrm{mN}/\mathrm{m}$, $K_A=0.44\mathrm{N}/\mathrm{m}$, $\beta =0.49$). The inset shows the time evolution of the contour. (f) Varying the compression velocity does not impact the fitting results: $T_0=0.83\mathrm{mN}/\mathrm{m}$, $K_A=0.39\mathrm{N}/\mathrm{m}$, $\beta =0.42$ for a MDCK II cell compressed with $0.5\mu \mathrm{m}/\mathrm{s}$ and $T_0=0.57\mathrm{mN}/\mathrm{m}$, $K_A=0.24\mathrm{N}/\mathrm{m}$, $\beta =0.43$ for the same cell compressed at $5\mu \mathrm{m}/\mathrm{s}$. (g) Compression-relaxation curve of a MDCK II cell after blebbistatin treatment showing substantial softening ($v_0=0.5\mu \mathrm{m}/\mathrm{s}$). Blue line: Hertz fit ($E_0=62\mathrm{Pa}$, $\beta =0.41$). Red line: Evans fit ($T_0=0.02\mathrm{mN}/\mathrm{m}$, $K_A=0.002\mathrm{N}/\mathrm{m}$, $\beta =0.57$). (h) MDCK II cell after calyculin $A$ treatment. Blue line: Hertz fit ($E_0=1097\mathrm{Pa}$, $\beta =0.18$). Red line: Evans fit ($T_0=1.6\mathrm{mN}/\mathrm{m}$, $K_A=2.36\mathrm{N}/\mathrm{m}$, $\beta =0.4$).

Figure 2

(a) Indentation-retraction curve of a confluent MDCK II cell probed with a conical indenter and subject to fitting with the viscoelastic Evans model (red line) and Hertz model (blue line), respectively. The insets show a cross section of the cell and the computed shape after indentation, respectively (scale bar: $10\mu \mathrm{m}$). (b) Young’s modulus $E_0$ of confluent MDCK II cells obtained for different indenter geometries (cone, spheres with various diameters). (c),(d) Corresponding $T_0$ and $K_A$ obtained from fitting with the viscoelastic Evans model.

Figure 3

Power law exponents $\beta$ of MDCK II cells (circles) and fibroblasts (squares) as a function of prestress $T_0$ and area-compressibility modulus $K_A$ subject to different indenter geometries and drug treatments (see legend). MDCK confl.: confluent MDCK II cells deformed with conical and spherical indenters (light blue, data compiled from Fig. 2); Blebb: Blebbistatin; CytD: Cytochalasin D; CalA: Calyculin A; GDA: Glutardialdehyde; Dashed lines are fits illustrating the logarithmic dependencies [15].