To test the hypothesis that fiber buckling facilitates long range propagation of cell-induced displacements, we developed a model for a fibrous network (Fig. 19.3) that simulated buckling as a loss of stiffness in compression. To do so, we considered a piecewise stress–strain curve having a ratio of stiffness in compression to stiffness in tension given by the constant ρ. For ρ <1 (e.g., ρ ¼0.1 in Fig. 19.3b, solid line), the fibers lose resistance to compression, hence mimicking buckling. We compare this model to one with ρ ¼1 (Fig. 19.3b, dashed line), thus giving a comparison between buckling and no buckling. In preliminary simulations, we model the contractile cell as a circle contracting in fibrous matrix (Fig. 19.4a). When fiber buckling is present (ρ ¼0.1), displacements u propagate over a long range with a fit to distance from the circle r matching u r 0.35. In contrast, when fibers do not buckle (ρ ¼1), displacements decay as u r 0.80, which more closely matches the solution from linear elasticity u r 1. Therefore, we conclude that buckling is sufficient to induce displacements that propagate over a long range. Since fiber buckling appears to be sufficient for long distance alignment and propagation of fiber displacements, it may allow the cell to mechanically probe its distant environment. Mechanical guidance cues are not undocumented, and cells have previously been shown to migrate toward increasing stiffness [4]. To ascertain if cells can leverage this basic property of fiber buckling to sense their environments, 3D stacks of fluorescently labeled cells and fibrin matrix were imaged simultaneously. As expected, cell-cell mechanical interactions were observed through matrix fiber remodeling between pairs cells (Fig. 19.5). Dense bundles of aligned fibers stretched between the cell bodies, and appear to interact over a range of 100μm. This range is approximately 5 times the diameter of a cell body, indicating that cells may sense one another through long range interactions facilitated by the nonlinear properties of the matrix. Fig. 19.3 Fiber network details. (a) Fiber network with coordination number C¼3. (b) Stress–strain curve for a linear fiber (ρ ¼1, dashed line) and a fiber that weakens in compression due to buckling (ρ ¼0.1, solid line). Adapted from Notbohm et al. [7] Fig. 19.4 Buckling causes displacements to propagate over a long range. (a) A contractile cell is modeled by a contracting circle of radius a. Normalized displacement u/a is shown for a model that simulates buckling (ρ ¼0.1). (b) Displacements are averaged around a circle of radius r about the center of the model for a simulation with a linear fiber (ρ ¼1, dotted line) and a buckling fiber (ρ ¼0.1, solid line). Plots shown here are for a coordination number of C ¼3. Adapted from Notbohm et al. [7] 19 Microbuckling of Fibrous Matrices Enables Long Range Cell Mechanosensing 139
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