Traction fields, moments, and strain energy that cells exert on their surroundings

TLDR

Adherent cells generate traction forces on their substrate, measurable by bead displacement on flexible gels. This paper presents an exact solution for computing traction fields from observed displacement data. The authors derive a Fourier‑space solution, offering two variants based on boundary traction assumptions and applying it to histamine‑stimulated airway smooth muscle cells. The method is computationally efficient and supplies explicit formulas to calculate contraction moments, traction principal axes, and strain energy from traction and displacement fields.

Abstract

Adherent cells exert tractions on their surroundings. These tractions can be measured by observing the displacements of beads embedded on a flexible gel substrate on which the cells are cultured. This paper presents an exact solution to the problem of computing the traction field from the observed displacement field. The solution rests on recasting the relationship between displacements and tractions into Fourier space, where the recovery of the traction field is especially simple. We present two subcases of the solution, depending on whether or not tractions outside the observed cell boundaries are set to be zero. The implementation is computationally efficient. We also give the solution for the traction field in a representative human airway smooth muscle cell contracted by treatment with histamine. Finally, we give explicit formulas for reducing the traction and displacement fields to contraction moments, the orientation of the principal axes of traction, and the strain energy imparted by the cell to the substrate.