Scaling of traction forces with the size of cohesive cell colonies

Abstract

To understand how the mechanical properties of tissues emerge from interactions of multiple cells, we measure traction stresses of cohesive colonies of 1-27 cells adherent to soft substrates. We find that traction stresses are generally localized at the periphery of the colony and the total traction force scales with the colony radius. For large colony sizes, the scaling appears to approach linear, suggesting the emergence of an apparent surface tension of the order of 10(-3) N/m. A simple model of the cell colony as a contractile elastic medium coupled to the substrate captures the spatial distribution of traction forces and the scaling of traction forces with the colony size.

FIG. 1

(color online). Traction stresses and strain energies for colonies of cohesive keratinoctyes. (a) Schematic of experimental setup (not to scale) with a cell colony adherent to an elastic substrate embedded with two dilute layers of fluorescent beads. (b), (d), (f) Distribution of traction stresses, σ_iz, and (c), (e), (g) strain energy, w, for a representative single cell, pair of cells, and colony of 12 cells. Traction stress distribution is overlaid on a DIC image (b) or images of immunostained cells (d), (f). Solid lines in (b), (c), (e), (g) mark cell boundaries. For clarity, only one-quarter of the calculated stresses are shown in (b), (d) and one-sixteenth of the stresses in (f). Scale bars represent 50 μm.

FIG. 2

Spatial distribution of strain energy for colonies of different size. Each solid curve represents a colony's average strain energy density as a function of distance from the edge of the colony,Δ. For clarity, the profiles are spaced vertically according to the size of the colony. Each profile terminates at the point where the inward erosion of the outer edge covers the entire area of the colony, at Δ ≈ R. The erosion proceeds in discrete steps of size δ, as illustrated in the inset.

FIG. 3

Mechanical output of keratinocyte colonies versus geometrical size. Total force transmitted to the substrate by the cell colonies, defined in Eq. (1), is plotted as a function of the equivalent radius of the colonies. The dashed line represents the scaling expected for surface tension, $\mathcal{F}\sim R$. The solid line shows a fit of the data to the minimal contractility model in Eq. (6).