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

Abstract

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.

The role of material structure and mechanical properties in cell–matrix interactions

“…Note that in both cases the elastic modulus of the material is the same. Note that this figure is for explanatory purposes only and ignores many other variables -for a full physical description of depth sensing please refer to Merkel et al 58 or Maloney et al 59 (b) The graph shows that the strain required to deform the gel a distance l increases according to a power law with decreasing gel thickness (A). The strain in the direction indicated is given by the percentage extension of the hypotenuse of an imaginary triangle with vertices marking (1) the focal adhesion at the gel surface prior to a hypothetical cell contraction (2) the focal adhesion at the gel surface after a hypothetical cell contraction and (3) the point of adherence of the gel to the underlying glass support directly below vertex (1).…”

Matrix elasticity, cytoskeletal forces and physics of the nucleus: how deeply do cells ‘feel’ outside and in?

“…glass in Fig. 1C), the long-range propagation of displacements will generally be affected; however, a thin matrix is subjectively thin only if the cell senses the rigid substrate (Maloney et al, 2008;Merkel et al, 2007). For example, cardiac fibroblasts cultured on synthetic gels of equal E that are either a few microns thin (Fig.…”

“…A formal theoretical basis for using cell-induced matrix deformations to calculate cell tractions relies on superposing the solution for a point force, f, acting at the surface of an isotropic substrate that is either of semi-infinite (Boussinesq, 1885) or finite (Maloney et al, 2008;Merkel et al, 2007) thickness. More complex geometries such as arbitrarily thin gels that are either flat or perhaps curved (per Fig.…”

Biological Activity of Fibronectin at the Cell-Material Interface

“…Since it has been shown that cells are able to sense the stiffness of the underlying substrate up to 1 µm depth, the underlying glass makes our substrates still stiffer from cells perspective. [51][52]. This would indicate that changes observed in cell response are not linked to stiffness of the substrate [53]; thus surface mobility is suggested as the determining factor in the cellular response.…”

“…Larger proteins typically bind stronger to the surface because of a larger contact area [52]. Hence, adsorption from protein mixtures is a selective phenomenon that leads to enrichment of the interface in particular proteins [11,13,19].…”