Quantifying forces generated by cells in three dimensions

Two-dimensional cell traction forces can be typically obtained by tracking substrate patterns and their deformation as the cell is actively exerting forces. From a measured displacement field, one can obtain quantitative information on the stresses exerted by the cell; however, this is only possible if several assumptions hold regarding the substrate:

· the material is isotropic;

· the material is elastic;

· the material is homogeneous;

· the material is linear (not necessary, though it greatly simplifies calculations)

· the material deforms in an affine way;

In three-dimensional tissue equivalents (cells embedded in 3D collagen matrix), support is provided by a collagen matrix, with typical concentrations of the order of 1mg/mL, mesh sizes in the micron range and fiber diameters of hundreds of nanometers. Several assumptions above do not hold in this case, at least not at the length-scale of a cell, which can be presumed to be the relevant one. It is unclear, though, how important each of these assumptions is. Therefore, we wish to compare quantitative stress/force measurements obtained through different models:

· homogeneous, continous, isotropic, matrix

· discrete fiber architecture

Figure 1: (left) Projections in Z of a cell contracting a fluorescently-labeled collagen matrix; (right) superimposed deformation grid calculated from image cross-correlation. Scale bar 30mm.

Collagen fluorescent labeling for quantitative network extraction

Many research groups use or reference confocal reflectance laser scanning microscopy as the imaging modality of choice for fibrillar collagen matrices. It has the advantage of requiring no label whatsoever and displays no photobleaching as a consequence. However, one cannot base any quantitative measurements on it, because the reflection response is strongly dependent on the orientation of fibers with respect to the orientation plane xy, as we have seen.

Figure 2: 6.4mm by 6.4mm projections comparing fluorescence and reflectance stacks of a 1mg/mL fluorescently labeled sample. (a) Maximum projection along Z of fluorescence imaging; (b) maximum projection along X of fluorescence imaging; (c) maximum projection along Z of reflectance imaging; (d) maximum projection along X of reflectance imaging; (e) maximum projection along Z of both imaging modalities, with fluorescence in red and reflectance in green; (f) maximum projection along X of both imaging modalities, same colors.

Fiber tracking of fluorescently-labeled collagen networks

For concentrations between 0.1mg/mL and 2.0mg/mL at 1:5 labeled to unlabeled ratio, our protocol provides not only beautiful images of these networks, but allows us to extract full 3D network information, which, theoretically, would provide the following information:

• fiber connectivity, i.e. coordination number

• segment lengths and total length per unit volume

• fiber persistence length

• mesh size

• qualitative idea of average fiber diameter

Applying this information to in vitro controlled sheared networks or to cell-induced collagen contraction, we can theoretically measure what is happening at the fiber level, whether it is stretching, rotating, bending or buckling. This will help better understand the forces exerted by cells in collagen and will also promote quantitative analysis of biopolymer network mechanics.

Figure 3 provides a qualitative comparison between a deconvolved fluorescent stack of 0.5mg/mL collagen and its 3D-extracted counterpart.

Figure 3: 25mm cube of a 0.5mg/mL fluorescently-labeled collagen network. (a) fully deconvolved three-dimensional volume imaging provided by Huygens software and (b) 3D-extracted network, with color-coding to indicate angle with the Z axis - red for vertical fiber segments and blue for horizontal ones.

Acknowledgements

We would like to thank the Swiss National Science Foundation for its financial support, as well as the American National Science Foundation, which provided funding through an IGERT biomechanics training grant. We would also like to thank the Center for Nanoscale Systems (CNS) for the use of confocal facilities.

The above work has been done in collaboration with the following people:

- L. Jawerth, K. Kasza, D. Weitz, Soft Condensed Matter Physics Laboratory, Engineering and Applied Sciences, Harvard University, Boston, MA, USA

- A. Kabla, L. Mahadevan, Applied Mathematics Laboratory, Engineering and Applied Sciences, Harvard University, Boston, MA, USA

- A. Stein, L. Sanders, Physics & Applied Mathematics, University of Michigan, Ann Arbor, MI, USA

- B. Hinz, J.-J. Meister, Laboratory of Cell Contractility, Life Sciences, Swiss Federal Institute of Technology, Lausanne, Switzerland.