# br plot its magnitude averaged over a slice

plot its magnitude averaged over a slice thickness of 510 mm above and below the cell (Fig. 4 A). This value is then normalized by the median stiffness tensor magni-tude within the imaged volume to quantify the local strain stiffening of the matrix. Note that the local matrix stiff-ening is much larger than the effective matrix stiffening that the cell ‘‘feels.’’ This is because stiffened matrix re-gions transmit cell deformations over large distances to-ward unstrained and thus softer matrix regions, which corresponds roughly to a serial arrangement of a stiff and a soft spring, with a total spring stiffness that is dominated by the soft spring.

No apparent structural changes can be observed in the collagen matrix for control cells, lamin-A-overexpressing cells, and 4-hydroperoxy Cyclophosphamide with beads (Fig. S8). However, in support of our hypothesis, we find a significantly higher stiffening of the collagen matrix around MDA-beads cells compared to MDA-control cells (Fig. 4 C). Interestingly, the matrix around MDA-lamA cells, despite a lower strain energy value, also shows some additional stiffening, attributable to the larger principal component force aligned with the ma-jor cell axis (Fig. 4 A).

Co´ndor et al.

FIGURE 3 Influence of cell rigidity on magni-tude and polarity of cell forces. (A) Total contrac-tility and (B) force polarity of control cells (blue, n ¼ 33), lam-A-overexpressing cells (green, n ¼ 33), and cells with 5-mm polystyrene beads (red, n ¼ 30) embedded in collagen gels. For all box plots, the central line indicates the median, the edges of the box denote the first and third quartiles, and the whiskers extend to the maximum and minimal value not considered as an outlier. Outliers are plotted as individual points. *p % 0.05, **p % 0.01, Student’s t-test assuming unequal variances including the outliers; n.s., not significant (p > 0.05). To see this figure in color, go online.

Effect of steric hindrance on cell morphology

We next explore the role of cell shape during cell invasion within 3-D collagen gels. From bright-field images stacks of cells embedded in 1.2 mg/mL collagen gels after 12 h of culture, we quantify the maximum projected cell length (‘‘major axis’’), the largest cell dimension perpendicular to the major axis (‘‘minor axis’’), and the cell aspect ratio (the ratio of major axis/minor axis).

MDA-lamA cells show similar morphology compared to MDA-control cells (Fig. 5 A). This indicates that the higher force polarity seen in MDA-lamA cells does not originate from a more elongated cell shape but instead from an altered distribution of tractions across the cell surface. We also find a significantly shortened major axis and consequently lower aspect ratio in MDA-beads cells, but this does not translate into a lower force polarity, again pointing to an altered distribution of tractions across the cell surface (Fig. 5, B–D). In line with this interpretation, we find that the contractility and cell aspect ratio of indi-

vidual cells under all three conditions show a weak nega-tive correlation (Fig. S1; Supporting Materials and Methods, Note 1), indicating that the more elongated cells tend to generate slightly smaller contractile forces, but this correlation does not reach statistical significance in our data.

DISCUSSION

In this study, we investigate how cells migrate through a collagen network and respond to changes in steric hin-drance that arise from altered cell mechanical properties. We increase steric hindrance either by stiffening the nu-cleus by overexpression of the nuclear envelope protein lamin A or by introducing 5-mm polystyrene beads into the cells that are larger than the average network pore size of 3.8 mm. Although both interventions may also cause secondary cellular responses that are difficult to pre-dict, we argue that the ability to measure cell-generated

FIGURE 4 Influence of contractile forces on elastic strain energy and stiffness of the collagen matrix. (A) Strain-stiffening maps of the collagen network around an embedded control cell (left), lam-A-overexpressing cell (middle), and a cell with a 5-mm polystyrene bead (right). (B) Strain en-ergy and (C) matrix strain stiffening of control cells (blue, n ¼ 33), lam-A-overexpressing cells (green, n ¼ 33), and cells with 5-mm polystyrene beads (red, n ¼ 30) embedded in collagen gels. *p % 0.05, ***p % 0.001, Student’s t-test assuming unequal variances including the outliers; n.s., not significant (p > 0.05). To see this figure in color, go online.

Cell Forces under Steric Hindrance

FIGURE 5 Influence of steric hindrance on cell morphology. (A) Maximal intensity projection image of a control cell (left), lam-A-overexpressing cell (middle), and a cell with a 5-mm polystyrene bead (right) embedded on a 1.2 mg/ml collagen gel. (B) Major cell axis length, (C) minor cell axis length, and (D) aspect ratio of control cells (blue, n ¼ 33), lam-A-overexpressing cells (green, n ¼ 33), and cells with 5-mm polystyrene beads (red, n ¼ 30) embedded in collagen gels. *p % 0.05, Student’s t-test assuming unequal variances including the outliers; n.s., not significant (p > 0.05). To see this figure in color, go online.