Toward Measuring the Mechanical Stresses Exerted by Branching Embryonic Airway Epithelial Explants in 3D Matrices of Matrigel

Abstract

Numerous organs in the bodies of animals, including the lung, kidney, and mammary gland, contain ramified networks of epithelial tubes. These structures arise during development via a process known as branching morphogenesis. Previous studies have shown that mechanical forces directly impact this process, but the patterns of mechanical stress exerted by branching embryonic epithelia are not well understood. This is, in part, owing to a lack of experimental tools. Traditional traction force microscopy assays rely on the use of compliant hydrogels with well-defined mechanical properties. Isolated embryonic epithelial explants, however, have only been shown to branch in three-dimensional matrices of reconstituted basement membrane protein, or Matrigel, a biomaterial with poorly characterized mechanical behavior, especially in the regime of large deformations. Here, to compute the traction stresses generated by branching epithelial explants, we quantified the finite-deformation constitutive behavior of gels of reconstituted basement membrane protein subjected to multi-axial mechanical loads. We then modified the mesenchyme-free assay for the ex vivo culture of isolated embryonic airway epithelial explants by suspending fluorescent microspheres within the surrounding gel and tracking their motion during culture. Surprisingly, the tracked bead motion was non-zero in regions of the gel far away from the explants, suggestive of passive swelling deformations within the matrix. To compute accurate traction stresses, these swelling deformations must be decomposed from those generated by the branching explants. We thus tracked the motion of beads suspended within cell-free matrices and quantified spatiotemporal patterns of gel swelling. Taken together, these passive swelling data can be combined with the measured mechanical properties of the gel to compute the traction forces exerted by intact embryonic epithelial explants.

Keywords: Computational modeling; Epithelial morphogenesis; Micro-mechanical testing; Reconstituted basement membrane protein; Traction force microscopy.

Figure 1:. Overview of mechanical testing setup.

(A) Schematic depicting the preparation of 3 mm-diameter cylindrical specimens of Matrigel. (B-C) Experimental setup for the multi-axial, finite-deformation mechanical tests. Briefly, a platen affixed to an actuator-controlled cantilever beam is used to apply a compressive force $F$ and/or a shear force $F_s$ to a cylindrical sample of Matrigel. A side-view camera, positioned outside the chamber, is used to observe the axial displacement $\delta$ and/or the shear displacement $k$.

Figure 2:. Unconfined compression tests of cylindrical Matrigel specimens.

(A) Representative time-lapse images of a Matrigel specimen deforming in response to a compressive axial force $F$. The sample has an initial height $L_0$ at $t=0$ s, which decreases by an axial displacement $\delta =L_0-L$, where $L$ is the deformed height of the sample. The white dashed line indicates the initial height of the sample. (B-D) Representative (B) $\delta$ vs. $t$, (C) $F$ vs. $t$, and (D) $F$ vs. $\delta$ plots for repeated tests using an individual Matrigel specimen. Mean ± s.d. is shown for 6 repeated measurements. (E) Schematic depicting how the Cauchy stress $\sigma$ and axial stretch ratio $\lambda$ are computed during an unconfined compression test. (F) Computed $\sigma$ vs. $\lambda$ plot for the individual specimen depicted in (B-D). Mean ± s.d. is shown for 6 repeated measurements. (G) Computed $\sigma$ vs. $\lambda$ plot for multiple specimens taken from an individual lot of Matrigel. Mean ± s.d. shown for n = 19 specimens.

Figure 3:. Different lots of Matrigel exhibit distinct mechanical properties.

(A) The analytical solution for the unconfined compression of an incompressible, neo-Hookean cylinder gives a explicit relation between the stress $\sigma$ and the axial stretch ratio $\lambda$, which depends on a single free parameter: the modulus $C$. (B) Nonlinear regression analysis of experimental $\sigma$ vs. $\lambda$ data from an individual lot of Matrigel. Mean ± s.d. shown for 6 repeated measurements. Dashed red line represents the regression curve. (C) Nonlinear regression analysis of $\sigma$ vs. $\lambda$ data from multiple different lots of Matrigel. (D) Comparison of the computed modulus $C$ for multiple different lots of Matrigel. Mean ± s.d. shown for n=19, n=19, and n=20 samples from Lots 1, 2, and 3, respectively. A one-way ANOVA with a Tukey post hoc test was used to determine significance between groups; *** p < 0.001.

Figure 4:. Predicting the mechanical behavior of Matrigel in response to multi-axial loads.

(A) Schematic depicting the unconfined compression and shear of a cylindrical Matrigel specimen. Once the sample is compressed by an axial force $F$, it is then subjected to a shear force $F_s$ that generates the shear displacement $k$. (B-C) Representative (B) displacement vs. $t$ and (C) force vs. $t$ plots for the compression and shear of an individual Matrigel specimen. Mean ± s.d. is shown for 6 repeated measurements. (D-E) Representative (D) $\sigma$ vs. $\lambda$ and (E) $F_s$ vs. $k$ plots for an individual Matrigel specimen. Mean ± s.d. is shown for 6 repeated measurements. (F-G) 3D finite element simulation of the multi-axial loading experiments. During the unconfined compression phase of the simulation, the axial stresses ${\sigma }_{zz}$ increase in magnitude while the shear stresses ${\tau }_{xz}$ remain close to zero. When a shear displacement $k$ is applied to the upper surface of the specimen, ${\tau }_{xz}$ also begins to increase and becomes non-uniformly distributed within the material. The predicted shear force $F_s$ is calculated by integrating ${\tau }_{xz}$ along the upper surface of the specimen. (H) Nonlinear regression analysis of $\sigma$ vs. $\lambda$ data from the unconfined compression phase of the experiment, which is used to compute the modulus $C$. Mean ± s.d. is shown for 6 repeated measurements. (I) Comparison between measured and predicted plots of $F_s$ vs. $k$. Mean ± s.d. is shown for 6 repeated measurements.

Figure 5:. Tracking bead motion around cultured embryonic airway epithelial explants in 3D Matrigel.

(A-B) Brightfield (A) and confocal fluorescence (B) images of a representative embryonic airway epithelial explant cultured in 3D Matrigel at 0 and 24 hr. The confocal fluorescence images capture the motion of fluorescent microspheres embedded within the 3D matrix. (C) Tracked motion of the fluorescent beads surrounding the explant shown in panels A and B. The largest computed displacements were in close proximity to the tissue, but significant non-zero bead motion (dashed white box) was observed far from the explant. Scale bars, 100 μm.

Figure 6:. Quantifying matrix swelling in cell-free domes of Matrigel.

(A) Schematic of experimental setup. Bead motion was tracked within 2D material slices of cell-free domes of Matrigel that had been incubated at 37°C for 24 hr. (B-C) Time-lapse confocal fluorescence images (B) of the beads embedded within a 2D material slice of a representative sample at 2, 10, 15, and 20 hr of culture. The tracked bead motion (C) revealed a radially symmetric pattern of gel swelling, with the highest bead displacements at the edge of the sample. The white dashed line indicates the periphery of the gel at $t=0$ hr. Scale bars, 1000 μm.

Figure 7:. Time-varying principal Lagrangian strain distributions within cell-free domes of Matrigel.

(A-B) Representative (A) first and (B) second principal Lagrangian strain distributions ($E_1$ and $E_2$, respectively) within a 2D material slice of a cell-free dome of Matrigel at 2, 10, 15, and 20 hr of culture. Scale bars, 1000 μm.

Figure 8:. Quantifying swelling deformations within 2D material slices in cell-free domes of Matrigel.

(A) Representative confocal image showing the fluorescent microspheres embedded within a cell-free dome of Matrigel. Swelling deformations were compared at locations near the center and edge of the sample. Scale bar, 1000 μm. (B-D) Representative plots of (B) $I_1-3$ vs. $t$, (C) $E_1$ and $E_2$ vs. $t$, and (D) the swelling anisotropy index $E_1/E_2$ vs. $t$ at the center and edge of the sample. (E-F) Comparison of swelling deformations over time at the center and edge of the gel, as quantified by (E) $I_1-3$ or (F) the swelling anisotropy index $E_1/E_2$. Mean ± s.d. is shown for n=9 experimental replicates, using gels fabricated with different lots of Matrigel. A two-way ANOVA with a Tukey post hoc test was used to determine significance between groups; ** p < 0.01; **** p < 0.0001.

Figure 9:. Overview of the methodology to compute the traction stresses exerted by cultured embryonic epithelial explants in 3D Matrigel.

The method involves three steps: (1) Measure the mechanical properties of an individual lot of Matrigel; (2) Track the motion of fiducial markers embedded within the 3D matrix around a cultured embryonic explant and quantify any swelling deformations far from the tissue; (3) Use the elastic component of the deformation to estimate the traction stresses exerted by the embryonic explant.