br are allowed to migrate in the collagen
are allowed to migrate in the collagen gels for 12 h. Sub-sequently, we image the cell morphology as well as the collagen network around AB-7-FUBAICA with confocal reflection microscopy before and after cell relaxation with cytocha-lasin-D (Fig. 2, A–D). Cellular forces are then computed (Fig. 2 F) from the local displacements field of the collagen matrix (Fig. 2 E) using a nonlinear, semiaffine, finite-element 3-D force reconstruction algorithm (14).
Force maps of cells from the three groups (Fig. 2) show inward-directed (contractile) forces that are highly polar-ized, but no obvious differences between the groups are visible (see Figs. S2–S7 for a complete set of the experi-ments). We therefore compute two robust scalar character-istics of the force field, namely cell contractility and force polarity (Fig. 3). Cell contractility measures the total magnitude of the projected force vectors in the direction of the force epicenter, which is the point at which the norm of the cross-products of the nodal forces and the vec-tors from their respective nodes to that point is minimal (14). Force polarity is the maximum (principal) dipole contractility divided by the total contractility. Thus, the force polarity quantifies the fraction of the contractile force that is oriented in a single direction. For a force dipole, the force polarity approaches unity, whereas for an isotropic spherical force field, the force polarity ap-proaches 1/3.
By contrast, force polarity is similar in MDA-control cells (0.53 5 0.01, mean 5 standard error) and MDA-beads cells
Taken together, these findings suggest that cells can compensate for an increased steric hindrance of the matrix by independently adjusting the magnitude and polarity of contractile forces.
Strain energy and matrix stiffening
We next ask whether these compensatory strategies prompt the cells to invest different levels of energy for deforming the matrix. Deformation energy is calculated as the total elastic energy stored within in the imaged biopolymer network and released within 30 min after cells are treated with cytochalasin-D. Compared to MDA-control cells, MDA-lamA cells invest a significantly (p < 0.05) lower strain energy, whereas MDA-beads cells invest similar levels of strain energy despite their higher contractile forces (Fig. 4 B). Although we cannot measure the total en-ergy expenditure of the cells, our finding of an equal or lower strain energy suggests that the generated forces are utilized more efficiently under conditions of higher steric hindrance.
MDA-lamA cells achieve this higher efficiency with a more polarized force generation and hence by reducing ‘‘wasteful’’ forces perpendicular to the migration direction.
Cell Forces under Steric Hindrance
FIGURE 2 3-D cell forces of MDA-MB 231 cells in collagen. (A) A bright-field image of an MDA-MD 231 breast carcinoma cell migrating within the collagen gel. (B) An overlay of confocal reflection images of collagen fibers before (red) and after (green) cell relaxation with cytocha-lasin-D. Enlarged section of two regions with small (C) and large (D) deformations. (E) 3-D matrix displacement field and (F) 3-D force density maps of a control cell (left), lam-A-overexpressing cell (middle), and a cell with a 5-mm polystyrene bead (right). Marker length and color indicate the magnitude of the displacement and force vectors. The bottom face shows a bright-field maximum in-tensity z-projection of the gel volume. Scale bars, 50 mm. To see this figure in color, go online.
In the case of MDA-beads cells, which have a similar force polarity but higher total forces compared to control cells, we hypothesize that the lower-than-expected strain energy may be a direct consequence of collagen fibers stiffening under higher forces. Fiber stiffening increases the alignment be-tween collagen fiber orientation and cell-generated forces and generally allows for a more efficient conversion of trac-tions to movements.
To quantify the effective stiffening of the collagen matrix, we compute the second derivative of the total strain energy in response to an infinitesimal extra deformation of the cell, which gives the apparent matrix stiffness that the cell ‘‘feels.’’ This matrix stiffness is computed under two condi-tions: first, considering the full nonlinear collagen behavior, and second, considering only the linear and the buckling behavior of the collagen fibers but without any strain stiff-ening. The strain-stiffening value is the ratio of both stiff-ness values. It quantifies the additional overall matrix stiffness due to strain stiffening of the matrix.
We also compute the local map of strain stiffening of the collagen network around cells. From the strain vector of every node, we compute the matrix stiffness tensor and