Single Cell Mechanics: Stress Stiffening and Kinematic Hardening

Cell mechanical properties are fundamental to the organism but remain poorly understood. We report a comprehensive phenomenological framework for the complex rheology of single fibroblast cells: a superposition of elastic stiffening and viscoplastic kinematic hardening. Despite the complexity of the living cell, its mechanical properties can be cast into simple, well-defined rules. Our results reveal the key role of crosslink slippage in determining mechanical cell strength and robustness.

Figure 1

A fibroblast adhering between two microplates. The cell has a well-controlled and simple geometry. It adheres to the microplates via biochemical linkers, which help define the state of the cell cytoskeleton. (a) Right after contact. Bar: $10\mu \mathrm{m}$. (b) After $\sim 20\min $ at $35°\mathrm{C}$, strongly adhering cells often adopt a concave shape. From the apparent diameter $D_0$ (arrows) we estimate the initial area $A_0≔\pi (D_0/2{)}^2$. (c) Under large stretch.

Figure 2

Loading and unloading at constant rate $\dot{L}$. (a) Imposed length $L$ and measured Lagrangian stress $F/A_0$ (where $A_0$ is the initial cell contact area) as a function of time. After each ramp, $F$ stabilizes at a nonzero value $\mathfrak{F}$ (dotted line). (b) $F/A_0$ as a function of $L/L_0$ (where $L_0$ is the steady zero-force-length) during loading and unloading. Black curve: $\dot{L}=1\mu \mathrm{m}/\mathrm{s}$. Gray curve: $\dot{L}=0.03\mu \mathrm{m}/\mathrm{s}$.

Figure 3

Small amplitude stiffening, large amplitude linearity. (a) $F$ and $L$ as a function of time. (b) $F/A_0$ as a function of $L/L_0$. For clarity only a few loops are shown. Gray curves: experimental data. Black curves: fit using Eqs. (1, 2, 3, 4, 5), with $\gamma =5$ and $\beta =0.25$. The dashed line highlights the linear relation between the average values. Inset: “Differential” modulus $|\Theta |$ from the oscillations as a function of average force $⟨F⟩$ for the data in (b). The modulus exhibits stress stiffening, following the master relation reported in [7].

Figure 4

Elastic/inelastic behavior. (a) Imposed length $L$ as a function of time. (b) $F/A_0$ as a function of $L/L_0$ for a given cell. Reversible (elastic) behavior upon direction reversal is observed only close to a previous turning point, as in C, E, H. The response becomes irreversible (inelastic) after a steady large deformation: at the turning points D, F, G, I, the $F(L)$ curve does not retrace its previous path. Between $F$ and $I$ the experiment is equivalent to C–F, but in the unloading direction. Since the response is equivalent, the sense of deformation is irrelevant. (c1) Another cell, probed at a rate $\dot{L}=0.1\mu \mathrm{m}/\mathrm{s}$. (c2) Same cell as c1 but at $\dot{L}=1\mu \mathrm{m}/\mathrm{s}$. (d1) Another cell, $0.1\mu \mathrm{m}/\mathrm{s}$. (d2) Same cell as d1, $1\mu \mathrm{m}/\mathrm{s}$. (e) Fit using Eqs. (1, 2, 3, 4, 5) (black curve) to the data shown in Fig. 4b (gray curve), with $\gamma =8$ and $\beta =0.24$.

Figure 5

Loading and unloading for several rates $\dot{L}$. (a) $F$ as a function of $L/L_0$ for a given cell, where $L_0$ is the initial length. Rates are: (1) $10\mathrm{nm}/\mathrm{s}$, 2) $30\mathrm{nm}/\mathrm{s}$, 3) $0.1\mu \mathrm{m}/\mathrm{s}$, 4) $1\mu \mathrm{m}/\mathrm{s}$. The overstress $\Delta F$ is defined as the extent of relaxation after unloading. (b) $\Delta F$ as a function of rate $\dot{L}$ for 7 different cells (symbols). The curves are fits to Eq. (3).

Figure 6

(a) Controlled length $L$ as a function of time. A single loading cycle with amplitude 30% is performed. (b) Force $F$ as a function of length $L$. “Normal”: before adding glutaraldehyde. “Fixed”: in presence of glutaraldehyde 0.1%. The dotted line is a fit to Eq. (2). (c) The derivative $dF/dL$ of the fixed curve (black line) plotted against the differential master-relation (gray dots, as discussed in Ref. 7).