Measurement of mechanical tractions exerted by cells in three-dimensional matrices

Abstract

Quantitative measurements of cell-generated forces have heretofore required that cells be cultured on two-dimensional substrates. We describe a technique to quantitatively measure three-dimensional traction forces exerted by cells fully encapsulated in well-defined elastic hydrogel matrices. Using this approach we measured traction forces for several cell types in various contexts and revealed patterns of force generation attributable to morphologically distinct regions of cells as they extend into the surrounding matrix.

Figure 1

Cell-induced hydrogel deformations and construction of a discretized Green’s function. (a) Volume rendering of a GFP-expressing NIH 3T3 fibroblast (green) spreading into a 3D hydrogel containing fluorescent beads (red). Scale bar = 50 µm, 10 µm (inset). (b) Surface mesh of the cell. Scale bar = 50 µm. Bead displacement trajectories are mapped and color coded by magnitude. (c) 2D slices through the volume showing the magnitude of the peak principal strain in the hydrogel surrounding the cell. (d) Plots of bead displacements and hydrogel strain as a function of distance from the cell surface. (e) Schematic outlining the use of the finite element method to reconstruct the Green’s function. Surface traction (T), applied to the highlighted facet, induces displacements of the surrounding beads (g_ij,, inset). When repeated over all facets and beads, these relationships describe a discretized Green’s function that can be used to calculate the tractions applied by the cell (Supplementary Note 2). The subscript indices of T and g represent the Cartesian components of the bead displacement in direction i in response of an applied surface traction in direction j.

Figure 2

Measurement of tractions exerted by live cells. (a) Contour plot of the tractions (magnitude) exerted by the cell. (b) Magnified sections outlined in a showing the individual traction vectors on each facet. (c) Plot of the traction magnitudes as a function of the normalized distance from the center of mass (COM) of the cell. This normalized distance is approximately 1 for the most spread regions (such as tips of long slender extensions) and approximately 0 for the central cell body. (d) Mean traction at a given angle for cells encapsulated in 978 ± 228 Pa hydrogels. The angle (θ) was computed between the traction vector (T) and the position vector (r) of the cell facet with respect to the center of mass of the cell (inset). Plots shown are for least spread (0.0–0.2), moderately spread (0.4–0.6) and most spread (0.8–1.0) regions of cells. Data from c and d are from n = 12 cells from each condition.

Figure 3

Measurement of dynamic tractions exerted by spreading cells. (a) Contour plot of the tractions (magnitude) exerted by a cell as it invades into the surrounding hydrogel. Stabile and invading extensions are labeled i and ii, respectively. Scale bar = 20 µm. (b) Tractions exerted by extensions labeled in a as a function of distance from the center of mass of the cell.