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.