Mechanosensing of substrate thickness

Structure and function of the adherent cell depend in a crucial way on its microenvironment, including the stiffness of its substrate. It is often asserted that substrate thickness (as opposed to stiffness) plays a negligible role and therefore may be considered semi-infinite. This assertion has been recently challenged, but the characteristic length scale to consider in this regard is poorly understood. We show here that this characteristic length scale is the lateral cell size. As substrate thickness approaches the lateral dimension of the cell, the apparent stiffness of the substrate is amplified to levels much greater than the intrinsic stiffness of the substrate. This change in apparent stiffness is readily sensed by the cell, leading to changes in cell spreading area, stiffness, and contractile forces. In contrast to these responses that occur over the length of the cell, mechanosensing around an isolated point force is influenced greatly by intrinsic substrate stiffness but to a negligible extent by substrate thickness. We conclude that mechanosensing of substrate thickness is dominated in large part by traction forces spread over the lateral dimension of the cell.

Figure 1

Effect of substrate thickness, $h$, on the apparent stiffness of the gel; $q$ is the magnitude of the Fourier wave number (effectively the inverse of lateral cell dimension). The stiffness amplification factor, $S$, is the ratio of tractions obtained from a finite-thickness formulation [Eq. (1)] to that obtained from the Boussinesq solution [Eq. (5)]. The black line depicts stiffness amplification calculated from the traction components in the same orientation as the displacement wave vector (e.g., the $x$ component of tractions when the displacements vary sinusoidally in the $x$ direction). The gray line depicts stiffness amplification for tractions orthogonal to the displacement wave vector (e.g., the $x$ component of tractions when the displacements vary sinusoidally in the $y$ direction). The dotted lines are the asymptotic behaviors for thin substrates; these correspond to simple shear between two parallel plates.

Figure 2

Decay of displacements, $u_x$, away from the point force, $F_x$, along the $x$ axis. Close to the point force $\left(x/h⪡1\right)$ the displacements on a finite-thickness substrate decay with the same rate as the displacements on a semi-infinite substrate (line without markers). Beyond $x/h=1$, these displacements decay significantly faster. The nondimensionalized displacement $u_{x}hE/F_x$ is solely a function of $x/h$, with $E$ as the intrinsic stiffness of gel.

Figure 3

Cell spreading of HFL fibroblast cells on different thicknesses of polyacrylamide gel. (a) Average cell area as a function of substrate thickness, with cells plated either on wells of uniform thickness or on a wedge. Error bars are standard deviations on more than 400 cells. (b) Schematic diagram of wedge-shaped gel making protocol. (c) F-actin staining of HFL cells on thin region $\left(50\mu \text{m}\right)$ of wedge and (d) F-actin staining of HFL cells on thick region $\left(400\mu \text{m}\right)$ of wedge (scale $\text{bar}=100\mu \text{m}$).

Figure 4

Elastic modulus, $g^{\prime }$ $\left(\text{mean}\pm \text{SE}\right)$, measured at 0.75 Hz of HFL fibroblast cells on different thicknesses of polyacrylamide gel. Zero thickness corresponds to measurements done with cells plated on glass. Error bars are standard deviations; $n=65–164$ beads attached on a similar number of cells.

Figure 5

(a) Net contractile moment and (b) net strain energy of cells plated on gels with different thicknesses. Computations done by unconstrained FTTC. Note the sharp increase in contractility and strain energy of the cells on $50\mu \text{m}$ thickness gel. Error bars are standard errors, $n=5–10$ cells.