Contractile Stresses in Cohesive Cell Layers on Finite-Thickness Substrates

Using a minimal model of cells or cohesive cell layers as continuum active elastic media, we examine the effect of substrate thickness and stiffness on traction forces exerted by strongly adhering cells. We obtain a simple expression for the length scale controlling the spatial variation of stresses in terms of cell and substrate parameters that describes the crossover between the thin and thick substrate limits. Our model is an important step towards a unified theoretical description of the dependence of traction forces on cell or colony size, acto-myosin contractility, substrate depth and stiffness, and strength of focal adhesions and makes experimentally testable predictions.

Figure 1

Schematic of a cell layer of lateral extent $L$ and thickness $h_c\ll L$ adhering to a substrate of thickness $h_s$. The build up of contractile stress $\sigma$ in the cell layer is indicated by the color map, while the traction stresses in the substrate are shown as vectors (blue online). The spatial variation of both traction and cellular stresses in the lateral ($x$) direction are characterized by the length scale ${\ell }_p$, referred to as the penetration length.

Figure 2

Top: Schematic of a contractile cell adhering to a soft substrate (left) and effective spring constant $k_{\mathrm{eff}}$ versus cellular strain $\Delta \ell$, showing strain stiffening (right). Bottom: Cell contraction $\Delta \ell$ (solid blue line) and traction force $F_T$ (red dashed line) vs substrate stiffness (left) for $h_s=10\mu \mathrm{m}$ and as a function of substrate thickness (right) for $E_s=500\mathrm{Pa}$. Other parameters: $F_A=10\mathrm{nN}$, $k_c=1\mathrm{nN}/\mu \mathrm{m}$, $k_a=2.5\mathrm{nN}/\mu \mathrm{m}$, $E_s=1\mathrm{kPa}$, $h_s=10\mu \mathrm{m}$, ${\ell }_{c0}=10\mu \mathrm{m}$, ${\nu }_s=0.4$.

Figure 3

Internal stress $\sigma (x)/{\sigma }_a$ (top left), substrate displacement $u^s(x)$ (top right), and traction stress $T(x)/{\sigma }_a$ (bottom) vs position $x$ along the cell layer for $E_s=500\mathrm{Pa}$ (solid, blue), 50 Pa (dotted, green), and 10 Pa (dashed, red). The vertical dashed lines in the top right frame denote the cell layer’s edges. Inset (bottom right): magnitude of contractile moment $|\mathcal{P}|$ vs $E_s/B_c$. Inset (bottom left): $|\mathcal{P}|$ as a function of substrate thickness for $E_s=10\mathrm{Pa}$. Other parameters: $B_c=1\mathrm{kPa}$, $h_c=0.1\mu \mathrm{m}$, ${\ell }_a=6.3\mu \mathrm{m}$, $L=100\mu \mathrm{m}$, ${\nu }_s=0.4$.