Influence of finite thickness and stiffness on cellular adhesion-induced deformation of compliant substrata

Thin, mechanically compliant coatings commonly serve as substrata for adherent cells in cell biology and biophysics studies, biological engineering applications, and biomedical device design. The deformation of such a coating at the cell-substratum interface defines the link between cellular traction, substratum stiffness, and the chemomechanical feedback mechanisms responsible for cellular mechanosensitivity. Here we apply elasticity theory to investigate how this deformation is affected by the finite thickness of such a cell substratum. The model idealizes a cellular adhesion site (e.g., a focal adhesion) as a circular area of uniform tangential traction, and compares the deformation of a compliant semi-infinite material to that of a coating of the same material supported by a rigid base. Two parameters are identified and considered: center displacement (as a measure of adhesion site displacement) and normal strain gradient (as a measure of adhesion site distortion). The attenuation of these parameters provides two measures for the influence of a finite coating thickness and underlying rigid base on cell-mediated deformation of the compliant substratum. A dimensionless term in the resulting solutions connects the coating thickness to the characteristic size of the adhesion sites. This relation, and calculations of the minimum thickness at which the rigid base is practically undetectable by an adherent cell, are supported by existing experimental literature and our observations of the projected area of fibroblasts adhered to polyacrylamide hydrogel coatings with various thicknesses atop relatively rigid glass. The model thus provides a tool for estimating the effective stiffness sensed by a cell attached to a compliant coating. We also identify and consider conceptualizations of critical thickness, or minimum suitable thickness for an application, which depend on both the frame of reference and the cell behavior of interest. The appropriate usage of different definitions resolves the disparity in values reported in the literature. Finally, the distinction between adhesion site displacement and distortion noted in this model could be useful in designing substrata to elucidate the controlling mechanisms of cellular mechanosensing.

Figure 1

Epifluorescence optical micrograph (negative image) of 3T3 fibroblasts, cultured on glass and fixed and stained for the focal adhesion adaptor protein vinculin (dark areas are the antivinculin antibody). Discrete vinculin features, often appearing at the periphery of the cells, correspond to adhesion sites between the cell and the adjacent surface. Scale $\text{bar}=50\mu \mathrm{m}$.

Figure 2

(Color online) Schematic of adherent cell attached to (a) a half space, or semi-infinite substratum and (b) a coating of thickness $h$ of the same material grafted to a rigid base. Adherent cells are contractile, exerting tangential traction on adhesion sites, here idealized as circular areas. The circular surface outlines indicate the undeformed and deformed positions and boundaries of the adhesion sites. Traction-induced deformation is attenuated by the presence of the rigid base; the displacements $u_B$ and $u_C$, respectively, correspond to the Boussinesq, or semi-infinite substratum, solution and the finite coating solution, as described in the text. Note that the coating covers the rigid base and serves as a substratum for the adherent cell.

Figure 3

(Color online) Displacement and distortion of circular area of applied tangential traction with increasing shear-stress-to-substratum-stiffness ratio; substratum Poisson’s ratio $\nu =0.5$. The deformation is calculated for a tangential traction $T$ and substratum shear modulus $\mu$ or Young’s elastic modulus $E$ by using equations derived in the text.

Figure 4

Schematic for integrating the deflection due to a tangential traction in the $x$ direction around an arbitrary interior point $(x,y)$. The adhesion site is idealized as a circular area with radius $a$.

Figure 5

(Color online) Normalized (a) surface displacement $u_B\mu ∕Ta$, (b) normal strain ${\epsilon }_{xx}\mu ∕T$, and (c) distortion (normal strain gradient) $(\mu ∕T)d{\epsilon }_{xx}∕dk$ plotted as functions of $k$, the normalized radial distance to the adhesion site center, for three values of substratum Poisson’s ratio $\nu$ (0, 0.25, 0.5) along the $x$ axis $(\theta =0)$. A discontinuity in strain and its derivative exists at $k=1$ (or $r=a$); we therefore focus on quantifying effects at the center of the circle.

Figure 6

(Color online) (a) Normalized coefficient $U_1(h∕a)$ associated with the center displacement of a circular adhesion site of radius $a$ due to the presence of an underlying rigid base at depth $h$ as a function of $h∕a$ for three values of substratum Poisson’s ratio $\nu$ (0, 0.25, 0.5). (b) Normalized coefficient $U_2(h∕a)$ associated with the distortion of the same circular adhesion site. As the coating thickness $h\to \infty$, $U_1$ and $U_2$ approach unity, and the displacement and distortion approach that of the semi-infinite region solution. As $h$ approaches zero, the displacement and distortion approach zero, representing complete attenuation of deformation due to the constraint of the rigid base.

Figure 7

Schematic for integration of the deflection at an arbitrary depth under the center of a circular area of tangential traction in the $x$ direction.

Figure 8

(Color online) Exact numerical evaluation of $U_1(h∕a)$ in Eq. (20) compared to approximation $U_1^{\star }(z∕a)$ in Eq. (34); substratum Poisson’s ratio $\nu =0.5$ for both calculations.

Figure 9

(Color online) (a) Cell area $A$ for 3T3 fibroblasts adhered to polyacrylamide gels with Young’s elastic modulus $E=5.6\mathrm{kPa}$ and a range of thicknesses fabricated on glass. The reduction in thickness of a compliant coating on an underlying rigid base results in increased effective stiffness and an increase in spread cell area. Error bars represent standard error $(n⩾47)$. Inset: phase contrast photographs of cells adhered to relatively thin and thick coatings of the same formulation of polyacrylamide (scale $\text{bars}=50\mu \mathrm{m})$. (b) The same data plotted as a function of effective Young’s elastic modulus (calculated by dividing the actual Young’s elastic modulus by the deformation function $U_1$ or $U_2$) for two assumed adhesion site radii, 1 and $2\mu \mathrm{m}$, and compared to Solon et al.’s measurements of fibroblast area on relatively thick polyacrylamide gels [33]. In (b), cell area error bars are omitted and the predictions from $U_1$ and $U_2$ are slightly offset vertically for clarity.

Figure 10

(Color online) Open symbols are cell area measurements on $70\mu \mathrm{m}$ (squares) and $500\mathrm{nm}$ (circles) compliant polyacrylamide (PA) hydrogel coatings as reported by Engler et al. [36]; data and vertical error bars are as published in [36]. Closed symbols are the result of applying our model’s functions $U_1$ and $U_2$ to the $70\mu \mathrm{m}$ data to predict cell area on a $500\mathrm{nm}$ film. The predictions from $U_1$ and $U_2$ are slightly offset vertically for clarity, and the horizontal range of $U_1$ and $U_2$ is due to the range of assumed adhesion site radii $a$ from $1\text{to}2\mu \mathrm{m}$.