Substrate Contributions in Elastic Pillar Arrays: Correction of Cellular Force Measurements

Abstract

The generation of tensile forces by cells is crucial for adhesion, migration, and cell to cell communication. The magnitude and direction of attachment forces are routinely measured by the deflection of elastic pillars on which the cells have been cultured. As the experimental calibration of the pillar spring constant is tedious, many studies use bulk material properties plus pillar dimensions for the force calculations and consider only bending and shear of the pillar. In this paper we show that all models that neglect the elastic substrate beneath the pillar systematically overestimate the applied forces, typically by more than 20%. Using finite element simulations we find that the elastic substrate accounts for 10-35% of the total deflection at the pillar top. The additional contribution arises from substrate warping at the pillar base and a consequent tilting of the pillar axis. The theoretical findings were verified by experiments with a macroscopic pillar model. We derive an analytical formula which can be used to correct the force calculations for given material properties and pillar aspect ratio. We further investigate the substrate-mediated crosstalk between pillars. We find that the force-loaded pillar acts as a force dipole and the coupling of a second pillar can be described in the framework of its dipole field. The crosstalk diminishes with the third power of the pillar-to-pillar distance and is under the detection limit for most practical cases. Our results highlight the importance of correcting for the systematic errors when comparing cellular forces that were derived from pillar arrays with different dimensions.