Theory of deformable substrates for cell motility studies.

Linear theory is used to relate the tractions F applied by a cell to the resulting deformation of fluid, viscoelastic, or solid substrates. The theory is used to fit data in which the motion of a fluid surface in the neighborhood of a motile keratocyte is visualized with the aid of embedded beads. The data are best fit by modeling the surface layer as a two-dimensional, nearly incompressible fluid. The data favor this model over another plausible model, the planar free boundary of a three-dimensional fluid. In the resulting diagrams for the distribution of F, it is found that both curl F and div F are concentrated in the lateral extrema of the lamellipodium. In a second investigation, a nonlinear theory of weak wrinkles in a solid substrate is proposed. The in-plane stress tensor plays the role of a metric. Compression wrinkles are found in regions where this metric is negative definite. Tension wrinkles arise, in linear approximation, at points on the boundary between positive definite and indefinite regions, and are conjectured to be stabilized by nonlinear effects. Data for the wrinkles that would be produced by keratocyte traction are computed, and these agree qualitatively with observed keratocyte wrinkles.