Mechanically patterning the embryonic airway epithelium

Significance

During branching morphogenesis of the developing lung, it is generally thought that a spatial template of biochemical cues determines the airway branching pattern. Here, however, we demonstrate that physical mechanisms can control the pattern of airway branching. Using a combination of 3D culture experiments and theoretical modeling, we show that a growth-induced physical instability initiates the formation of new epithelial branches. Tuning epithelial growth rate controls the dominant wavelength of the instability, and thereby the branching pattern. These findings emphasize the role of mechanical forces during morphogenesis and indicate that lung development is not a closed genetic system. Physical cues also regulate the spatially patterned cell behaviors that underlie organ assembly in the embryo.

Abstract

Collections of cells must be patterned spatially during embryonic development to generate the intricate architectures of mature tissues. In several cases, including the formation of the branched airways of the lung, reciprocal signaling between an epithelium and its surrounding mesenchyme helps generate these spatial patterns. Several molecular signals are thought to interact via reaction-diffusion kinetics to create distinct biochemical patterns, which act as molecular precursors to actual, physical patterns of biological structure and function. Here, however, we show that purely physical mechanisms can drive spatial patterning within embryonic epithelia. Specifically, we find that a growth-induced physical instability defines the relative locations of branches within the developing murine airway epithelium in the absence of mesenchyme. The dominant wavelength of this instability determines the branching pattern and is controlled by epithelial growth rates. These data suggest that physical mechanisms can create the biological patterns that underlie tissue morphogenesis in the embryo.

Space-filling, branched networks form the basic architecture of several organs, including the lung, kidney, and mammary gland. In the developing embryo, these complex structures originate as simple epithelial tubes. To form a ramified network, the initial tubular geometry is molded by a series of branching events, patterned in both space and time (1–3). In most cases, branching involves reciprocal signaling between adjacent tissues (4, 5), but it remains unclear how the locations of new branches are determined.

Airway branching is highly stereotyped in the developing mouse lung (6) and regulated in part by fibroblast growth factors (FGFs) (7). New epithelial branches are thought to emerge at locations adjacent to a prepattern of focal expression of FGF10 in the neighboring mesenchyme (8, 9) (Fig. 1A), a molecular mechanism with remarkable similarity to the induction of Drosophila tracheal branching by the FGF homolog Branchless (10). Moreover, the prepattern of FGF10 is regulated by reciprocal feedback between core signaling pathways, including those downstream of sonic hedgehog, bone morphogenetic protein, Wnt, and Notch (5, 11–13). These molecular signals are thought to interact via a reaction-diffusion mechanism (14, 15) to generate the spatial template of FGF10 (16) and are typically assumed to be sufficient to pattern branch locations along the airway epithelium (5, 10). This rich molecular description, however, overlooks the role that mechanical cues might play in establishing biological patterns. The pattern of villi in the developing gut, for instance, is determined by physical buckling (17, 18).

Fig. 1.

Morphodynamics of mesenchyme-free branching. Schematic of FGF prepattern in (A) intact mouse lung and (B) during mesenchyme-free culture. (C) Schematic of mesenchyme-free culture assay. (D) Time-lapse bright-field images of mesenchyme-free epithelial branching. (Scale bars, 200 µm.) (E) Epithelial branches form simultaneously with characteristic wavelength $\lambda$. Traced contours of epithelium during (F) branch initiation and (G) branch elongation. (H) Epithelial perimeter over time for a representative explant.

When the mesenchyme is removed, thus disrupting reciprocal signaling, isolated epithelia still branch in response to FGF1 or FGF10 (19, 20). In these culture models, the exogenous growth factors are present ubiquitously, with no apparent spatial pattern (Fig. 1B). In the absence of a mesenchymal prepattern of FGF, it is unclear how the epithelial branching pattern is specified. These data suggest the requirement for additional, nonchemically templated patterning mechanisms.

Results

Epithelial Branches Form Simultaneously with a Characteristic Wavelength.

To determine the dynamics of mesenchyme-free branching, we performed time-lapse imaging analysis of epithelial explants and quantified the resulting kinematics. Mesenchyme was removed from embryonic day 12.5 mouse lungs, and isolated epithelia were embedded in 3D gels of reconstituted basement membrane protein (Matrigel) (Fig. 1C). Experiments using mice transgenic for GFP downstream of the vimentin gene promoter confirmed complete removal of the mesenchyme (Fig. S1). As the cultured epithelium grew and expanded, new branches formed simultaneously (Fig. 1D and Movie S1), casting the epithelium into a folded geometry with characteristic wavelength, $\lambda$, of ∼100 µm (Fig. 1 E and F). These branches then grew outward, extending further into the surrounding gel (Fig. 1G). During this process, the perimeter of the branching epithelium (Fig. 1 F and G) increased in time as the epithelium expanded (Fig. 1H).

Fig. S1.

Epithelial explants isolated from vim-GFP mice at E12.5. Bright-field and fluorescent images of explants (A and B) before and (C and D) after manual removal of the mesenchyme. Most denuded explants showed no fluorescence, indicating a lack of mesenchymal cells. (Scale bars, 200 µm.)

Spatial Patterns of Proliferation Do Not Appear Until Branches Have Already Formed.

The embryonic airway epithelium is highly proliferative in vivo (21), and it is often assumed that new branches result from localized growth (22). We therefore quantified 3D patterns of proliferation within mesenchyme-free explants (Fig. 2 A and B) and reconstructed the epithelial surface (Fig. 2C) using confocal images of immunofluorescence staining for E-cadherin (Movie S2). We then computed the mean curvature, $\kappa$, along this reconstructed surface and compared the distribution of mean curvature to that of cell proliferation (Fig. 2D). Mean curvature was highest in the distal tips of extending branches, coincident with elevated levels of proliferation (20, 23, 24) (Fig. 2D).

Fig. 2.

Patterns of cell proliferation during mesenchyme-free branching. (A) Mesenchyme-free explant after 0 h and 40 h of culture. (Scale bars, 100 µm.) (B) Three-dimensional reconstruction of proliferation assessed via EdU incorporation (green). Airway epithelium colabeled by immunofluorescence staining for E-cadherin (red). (Scale bar, 200 µm.) (C) Three-dimensional surface reconstruction of characteristic epithelial geometry after branch initiation. (D) Plots of both mean curvature and EdU staining after branch initiation. (E) Mesenchyme-free explant after 16 h of culture (EdU, green). (Scale bar, 100 μm.) (F) Three-dimensional surface reconstruction of characteristic unbranched geometry. (Scale bar, 100 µm.) (G and H) Plots of both (G) mean curvature and (H) EdU staining before branch initiation.

We applied a similar analysis to mesenchyme-free epithelia before the initiation of branching (Fig. 2 E–G). In these explants, variations in surface curvature were more attenuated, and mean curvature and cell proliferation were not spatially correlated (Fig. 2G). These data indicate that distinct patterns of cell proliferation do not emerge until branches have already formed, consistent with previous findings (23).

Because new branches form simultaneously, and patterns of proliferation are relatively uniform before the onset of branching (Fig. 2G), we hypothesized that a growth-induced mechanical instability might drive the observed epithelial folding, similar to how a beam on an elastic foundation buckles under in-plane compressive loading (25). At a critical compressive load, the initially straight beam is no longer stable and spontaneously develops a buckled, wave-like morphology.

Modeling Constrained Epithelial Growth in an Elastic Gel.

To test the plausibility of this hypothesis, we constructed a simple physical model for the proliferating airway epithelium supported by an elastic foundation (i.e., the surrounding gel) (Fig. 3A). As a first approximation, we ignored the effects of initial curvature and modeled the epithelium as an infinite flat layer (see SI Text for a more detailed description of the mathematical model). Epithelial proliferation was included by decomposing the overall mechanical strain in the layer ($\mathit{ϵ}$) into a component due to growth (${\mathit{ϵ}}_g$) and a component due to elastic deformation (${\mathit{ϵ}}^{\ast }$) using the relation $\mathit{ϵ}={\mathit{ϵ}}^{\ast }+{\mathit{ϵ}}_g$ (Fig. S2). [This superposition of strains is a linearized version of the more general theory for finite volumetric growth (26) and is valid for small deformations, a condition that, as a first approximation, was also assumed here.] We then incorporated this framework for biological growth into the classic equation for buckling of an elastic layer supported by an elastic half-space (27, 28). Here, the compressive in-plane force ($N_g$) was induced by constrained growth in the proliferating epithelium (i.e., for ${\mathit{ϵ}}_g>0$, $N_g$ is compressive).

Fig. 3.

Tuning branch wavelength. (A and B) Simple linear model for a growing infinite sheet supported by a viscoelastic foundation. (C–H) Tuning branch wavelength experimentally by varying the concentration of Matrigel in the surrounding gel. Traced epithelial contours reveal branching dynamics. (Scale bars, 200 µm.) (I) Branch wavelength as a function of gel concentration. Shown are mean ± SD for four experimental replicates. (J) Epithelial expansion during branch formation. Shown are mean ± SD for four experimental replicates. Statistics computed using one-way ANOVA with Tukey’s post hoc test; **P < 0.01; ***P < 0.001.

Fig. S2.

Schematic depicting linear theory for biological growth. A 1D beam element grows from its reference length $L_0$ to a new length $L_g$ in a stress-free manner. This changes the zero-stress length from $L_0$ to $L_g$ and is described by the growth stretch ratio ${\mathrm{\Lambda }}_g$. Mechanical loads are then applied to the element at its new, stress-free length $L_g$. In its final deformed state, the element now has a length $L$. This second transformation is characterized by the elastic stretch ratio ${\mathrm{\Lambda }}^{\ast }$. If the overall change in length is given by $\mathrm{\Lambda }=L/L_0$, then $\mathrm{\Lambda }={\mathrm{\Lambda }}^{\ast }{\mathrm{\Lambda }}_g$. For small deformation, $\mathrm{\Lambda }=1+\mathit{ϵ}=\left(1+{\mathit{ϵ}}^{\ast }\right)\left(1+{\mathit{ϵ}}_g\right)$. Neglecting higher-order terms, this yields $\mathit{ϵ}={\mathit{ϵ}}^{\ast }+{\mathit{ϵ}}_g$.

Because spatial patterns of proliferation were not observed before branching (Fig. 2 E–G), we considered the case of uniform growth in the epithelium, ${\mathit{ϵ}}_g(x,y)={\mathit{ϵ}}_g^o$. In this case, the proliferating epithelium expands, reaches instability, and folds out-of-plane (Fig. 3B) with a dominant wavelength given by

${\lambda }_{c{r}_e}=2\pi h\sqrt[3]{B/3B_f}$, where $B=E/(1-{\nu }^2)$ and $B_f=E_f/(1-{\nu }_f^2)$ represent elastic moduli for the epithelium and surrounding gel, respectively, and depend on the Young’s modulus $E$ and Poisson’s ratio $\nu$ of each material. Thus, for the case of an elastic instability, the dominant wavelength is controlled by the relative difference in stiffness between the two layers.

Experimentally Testing the Elastic Instability Hypothesis.

The results of the elastic model suggested that uniform growth along the airway epithelium might be sufficient to produce branches of characteristic wavelength in the absence of a biochemical template, with the relative difference in stiffness between the epithelium and the gel specifying the locations of new branches. To test this mechanism experimentally, we cultured epithelial explants in different concentrations of Matrigel, thereby varying the mechanical properties of the gel (Fig. 3 C–H), which were measured using unconfined compression tests (Methods and Fig. S3). Based on the above relation for ${\lambda }_{c{r}_e}$, we would predict the dominant wavelength to decrease as gel stiffness increases. Indeed, we found that branch wavelengths varied as a function of gel concentration, with shorter wavelengths observed in explants cultured within higher concentrations of Matrigel (Fig. 3I). However, importantly, epithelial growth also depended on gel concentration: Epithelial contour perimeters increased more rapidly (Fig. 3J), branches formed at earlier time points (Fig. 3 C–H), and the rate of proliferation increased (Fig. S4) at higher concentrations of Matrigel.

Fig. S3.

Measuring the mechanical properties of the gel. (A) Schematic of unconfined compression tests. A PDMS mold was used to create specimens of defined cylindrical geometry, which were mechanically loaded with a 12-mm-diameter glass coverslip. Fluorescent microspheres (500 nm in diameter) were suspended within the specimen, before gelation, to observe the gel deformations after loading. (B and C) Representative image of cylindrical gel specimen (B) within the PDMS mold and (C) after mold removal. (D) Fluorescent image of gel specimen before and after loading. (Scale bar, 200 \x{03bc}m.) (E) Measured elastic modulus as a function of Matrigel concentration. Error bars indicate SD. Statistics computed using one-way ANOVA with Tukey’s post hoc test; *P < 0.05. These values are similar to those reported elsewhere for Matrigel (49, 50).

Fig. S4.

Quantifying epithelial proliferation rates at different concentrations of Matrigel via EdU incorporation. (A–C) Epithelial explants after 16 h of culture in gels consisting of (A) 25%, (B) 50%, and (C) 100% Matrigel. (Scale bars, 200 µm.) (D–F) Fluorescent images of EdU incorporation in explants cultured in gels consisting of (D) 25%, (E) 50%, and (F) 100% Matrigel. (Scale bars, 50 µm.) (G) Composite high-magnification images of EdU and Hoechst labeling. (Scale bar, 10 µm.) (H) Epithelial proliferation rate as a function of Matrigel concentration. The number of EdU-stained nuclei was normalized with respect to the cumulative area of the regions of interest analyzed within each explant. Error bars indicate SD. Statistics computed using one-way ANOVA with Tukey’s post hoc test; *P < 0.05; **P < 0.01.

In addition to extracellular matrix proteins, Matrigel also contains growth factors that can induce proliferation (29). To parse the relative roles of mechanics and proliferation, we used methylcellulose to alter the mechanical properties of Matrigel (Fig. 4A), while maintaining constant ligand density. Altering the stiffness of the gel (Fig. S5) did not significantly affect branch wavelengths (Fig. 4 A–D), suggesting that the mechanical properties of the foundation do not control branching and, therefore, that a purely elastic instability does not pattern the locations of new branches. The concentration-dependent differences in epithelial growth rates described above (Fig. 3J), however, suggested the possibility for time-dependent, viscoelastic effects.

Fig. 4.

Branch wavelengths are patterned by a growth-induced viscoelastic instability. (A–D) Varying the mechanical properties of the gel using methylcellulose. No differences in branch wavelength were observed between explants cultured in gels with methylcellulose (84 ± 8 µm) and those cultured in gels without (85 ± 9 µm). Data are mean ± SD for four experimental replicates. Statistics computed using Student’s t test; P = 0.82. (E–H) Tuning epithelial proliferation using different concentrations of FGF in 100% Matrigel. Data are mean ± SD for four experimental replicates. (Scale bars, 200 µm.) (I) Branch wavelength as a function of FGF concentration. Statistics computed using Student’s t test; **P < 0.01. (J) Epithelial expansion over time at different concentrations of FGF. Data are mean ± SD for four experimental replicates. (K) Schematic for the viscoelastic time history of folding model. Noisy (infinitesimal) fluctuations in the initial geometry of the epithelial layer $v_0\left(x\right)$ are decomposed into Fourier components. As the epithelium grows, the amplitudes of those components, which correspond to the dominant wavelength of the instability, are preferentially amplified in time. (L and M) In our simulations, epithelial folds of characteristic wavelength $\lambda$ emerge (with increasing time $t$) from the initial noise $v_0\left(x\right)$ (red curve). For increasing ${\mathit{ϵ}}_g$, we predict shorter branch wavelengths, as observed experimentally.

Fig. S5.

Altering the mechanical properties of the gel using methylcellulose. Measured elastic modulus of gels containing either 50% Matrigel or 0.5% methylcellulose and 50% Matrigel. Shown are mean ± SD for four experimental replicates. Statistics computed using Student’s t test; **P < 0.01.

Epithelial Branching Is Driven by a Growth-Induced, Viscoelastic Instability.

Returning to our mathematical model (Fig. 3 A and B), we thus included the effects of viscoelasticity in our governing equations (see SI Text for a more detailed description of the model). As a first approximation, we considered the simplest case of an elastic epithelial layer supported by a Maxwell-type viscoelastic foundation. In this case, the dominant wavelength (Fig. 3B) was not predicted to depend on the mechanical properties of the foundation, and was instead given by the relation

${\lambda }_{c{r}_v}=\pi h\sqrt{B/N_{g_{cr}}},$ where $B$ represents the elastic modulus of the epithelium and $N_{g_{cr}}$ represents the critical in-plane load, which depends linearly on the amount of growth, ${\mathit{ϵ}}_{g_{cr}}$. This relation predicts shorter branch wavelengths at higher proliferation rates, consistent with those observed when varying the concentration of Matrigel (Fig. 3 C–H). Also, importantly, this mechanism is consistent with our methylcellulose experiments, which suggested that branch wavelengths are independent of the mechanical properties of the gel.

To further test this mechanism experimentally, we cultured explants in gels supplemented with different concentrations of FGF (Fig. 4 E–H) to modulate epithelial proliferation (Fig. S6) independently of the mechanical properties of the gel. Consistent with the predictions of our model, branch wavelengths decreased (Fig. 4I) and the rate of epithelial expansion increased at higher concentrations of FGF (Fig. 4J). The pattern of branching could thus be tuned by varying the epithelial growth rate.

Fig. S6.

Quantifying epithelial proliferation rates at different concentrations of FGF in 100% Matrigel. (A–D) Epithelial explants after 16 h of culture in the presence of (A and C) 100 ng/mL and (B and D) 200 ng/mL FGF1. Explants were pulsed with EdU for 1 h before fixation. (A and B) Bright-field and (C and D) fluorescent images showing EdU incorporation at low magnification. (Scale bars, 250 µm.) (E and F) Explants were costained for (E) EdU and (F) Hoechst 33258 to quantify proliferation rates. (Scale bar, 50 µm.) (G) Composite high-magnification image of white boxed region shown in E and F. Hoechst 33258-labeled nuclei (grayscale) are shown overlayed with those positive for EdU incorporation (green). (Scale bar, 10 µm.) (H) Epithelial proliferation rates as a function of FGF concentration. The number of EdU-stained nuclei was normalized with respect to the cumulative area of the regions of interest analyzed within each explant. Error bars indicate SD. Statistics computed using Student’s t test; ***P < 0.001.

We then extended our viscoelastic model to predict how these epithelial folds might emerge from the initially unpatterned tissue. Because, in actuality, no epithelium is perfectly flat, the geometry of the undeformed tissue was taken to consist of random, infinitesimally small deflections $\left[v_0\left(x\right)\right]$, which could then be decomposed into a Fourier series with different components at different spatial frequencies (Fig. 4K). In the model, these irregularities were sufficient to initiate a folding instability in the growing epithelial layer, and regular folds of characteristic wavelength emerged spontaneously from the noisy initial conditions. For different values of ${\mathit{ϵ}}_g$, the increase in amplitude (with time $t$) of each Fourier component within $v_0\left(x\right)$ depended upon how closely the wavelength of each corresponded to the dominant wavelength of the instability ${\lambda }_{c{r}_v}$. Those components with wavelengths nearest to ${\lambda }_{c{r}_v}$ grew most quickly, and thus established the wave-like geometry of the epithelial layer (see SI Text for a more detailed description of the model). Our simulations predicted shorter branch wavelengths at increased values of ${\mathit{ϵ}}_g$, consistent with our experimental results (Fig. 4 L and M). The computed branch wavelengths were robust to variations in the initial irregularities in the epithelium. Taken together, these results indicate that a viscoelastic instability, driven by proliferation, defines the locations of epithelial branches in the absence of mesenchyme.

SI Text

Modeling Biological Growth.

To model the mechanics of mesenchyme-free branching, we constructed a theoretical model for an infinite epithelial layer growing on an underlying gel. As a first approximation, we assumed small deformations and ignored the effects of initial epithelial curvature. Biological growth was included by decomposing the overall deformation in the epithelium into a component due to growth and a component due to elastic deformation. Briefly, within this framework, a 1D tissue element of reference length $L_0$ is allowed to grow to a new “grown” length $L_g$, which alters the zero-stress state of the material, and is defined by the growth stretch ratio ${\mathrm{\Lambda }}_g=L_g/L_0$ (Fig. S2). Applied traction forces then deform this grown configuration into the current length $L$, which invests strain energy in the tissue. This elastic deformation is defined by the elastic stretch ratio ${\mathrm{\Lambda }}^{\ast }=L/L_g$. Thus, the overall deformation of the 1D element can be defined using the stretch ratio $\mathrm{\Lambda }=L/L_0={\mathrm{\Lambda }}^{\ast }{\mathrm{\Lambda }}_g$. If small deformations are assumed, mechanical strain can be represented as $\mathit{ϵ}=\mathrm{\Lambda }-1$, and the overall strain in the tissue $\mathit{ϵ}$ can be approximated by the superposition of the elastic strain ${\mathit{ϵ}}^{\ast }$ and the growth strain ${\mathit{ϵ}}_g$, because $\mathit{ϵ}={\mathrm{\Lambda }}^{\ast }{\mathrm{\Lambda }}_g-1=\left(1+{\mathit{ϵ}}^{\ast }\right)\left(1+{\mathit{ϵ}}_g\right)-1\cong {\mathit{ϵ}}^{\ast }+{\mathit{ϵ}}_g$.

Purely Elastic Model.

We first considered the case of a growing epithelial layer (with height $h$, width $b$, Young’s modulus $E$, Poisson’s ratio $\nu$) supported by an elastic foundation (Fig. 3A). We incorporated biological growth into the classic equation for the buckling of an infinite elastic layer supported by an elastic half-space (27, 28). Growth of the epithelium was constrained by deformations in the underlying foundation. The mechanical stresses depended only on the elastic strain ${\mathit{ϵ}}^{\ast }$ and were given by $\sigma =E\left(\mathit{ϵ}-{\mathit{ϵ}}_g\right)$. Assuming a sinusoidal deflection in the layer, the governing equation takes the form

$$\frac{E{h}^3}{12\left(1-{\nu }^2\right)}\frac{d^{4}v}{d{x}^4}+N_{g}h\frac{d^{2}v}{d{x}^2}+\gamma \frac{E_f}{2\left(1-{\nu }_f^2\right)}v=-\frac{d^2{M}_g}{d{x}^2},$$

where $v\left(x\right)$ is the deflection, $\gamma$ represents the wave number of the assumed sinusoidal deflection, $E_f$ and ${\nu }_f$ are the elastic modulus and Poisson’s ratio of the foundation, respectively, $N_g=bE{{\displaystyle \int }}_{-h/2}^{h/2}{\mathit{ϵ}}_g\left(x,y\right)\hspace{0.17em}dy$ is the in-plane force due to growth, and $M_g=bE{{\displaystyle \int }}_{-h/2}^{h/2}y{\mathit{ϵ}}_g\left(x,y\right)\hspace{0.17em}dy$ is the bending moment due to nonuniform growth across the thickness of the layer.

Because no spatial patterns of proliferation were apparent before branching (Fig. 2 E–G), we considered the case of uniform growth in the epithelium, ${\mathit{ϵ}}_g(x,y)={\mathit{ϵ}}_g^o$, which yielded $M_g=0$, and $N_g=bEh{\mathit{ϵ}}_g^o$. Our governing equation thus simplified to

$$\frac{E{h}^3}{12\left(1-{\nu }^2\right)}\frac{d^{4}v}{d{x}^4}+N_{g}h\frac{d^{2}v}{d{x}^2}+\gamma \frac{E_f}{2\left(1-{\nu }_f^2\right)}v=0.$$

Assuming solutions of the form $v\left(x\right)=v_0\cos \left(\gamma x\right)$, this gives the characteristic equation

$$N_g=\frac{B}{12}{\left(h\gamma \right)}^2+\frac{B_f}{2}\frac{1}{\left(h\gamma \right)},$$

where $B=E/(1-{\nu }^2)$ and $B_f=E_f/(1-{\nu }_f^2)$ represent elastic moduli for the epithelium and surrounding gel, respectively, and depend on the Young’s modulus $E$ and Poisson’s ratio $\nu$ of each material. The system thus reaches instability at a critical in-plane load (due to growth) of

$$N_{g_{cr}}=\frac{3}{4}{B}_f\sqrt[3]{\frac{B}{3B_f}}$$

and has a dominant wavelength of

$${\lambda }_{c{r}_e}=2\pi h\sqrt[3]{\frac{B}{3B_f}}.$$

Viscoelastic Model.

We extended this model to incorporate the effects of viscoelasticity. In this case, the mechanical properties of the epithelium and foundation were governed by the viscoelastic operators $Q\left(p\right)$ and $R\left(p\right)$, and $Q_f\left(p\right)$ and $R_f\left(p\right)$, respectively, where $p=\partial /\partial t$ (45). Using the correspondence principle (46), this yielded the following governing equation:

$$B\left(p\right)\frac{h^3}{12}\frac{d^{4}v}{d{x}^4}+N_{g}h\frac{d^{2}v}{d{x}^2}+\frac{\gamma B_f\left(p\right)}{2}v=0,$$

where $B\left(p\right)=4Q\left(p\right)\left[Q\left(p\right)+R\left(p\right)\right]/(2Q\left(p\right)+R\left(p\right))$, and$\hspace{0.17em}{B}_f\left(p\right)$ is defined similarly (28).

Assuming solutions of the form $v\left(x\right)=v_0\cos \left(\gamma x\right)$, this produced the characteristic equation

$$N_g=B\left(p\right)\frac{{\left(h\gamma \right)}^2}{12}+\frac{1}{2\left(h\gamma \right)}{B}_f\left(p\right).$$

The system reached instability at a critical in-plane force (due to growth) of

$$N_{g_{cr}}=\frac{3}{4}{B}_f\left(p\right)\sqrt[3]{\frac{B\left(p\right)}{3B_f\left(p\right)}}$$

and had a dominant wavelength given by

$${\lambda }_{cr}=2\pi h\sqrt[3]{\frac{B\left(p\right)}{3B_f\left(p\right)}}.$$

For simplicity, we considered a growing elastic layer supported by a Maxwell-type viscoelastic foundation. The foundation properties were given by $Q_f\left(p\right)=G_{f}p/(p+r_f)$ and $R_f\left(p\right)=\infty$, where $r_f=G_f/{\mu }_f$, $G_f$ is the shear modulus of the foundation, and ${\mu }_f$ is its effective viscosity. In this case,

$$B_f\left(p\right)=\frac{4G_{f}p}{p+r_f}.$$

For the elastic layer, the viscoelastic operators $Q\left(p\right)$ and $R\left(p\right)$ reduced to the Lamé constants ${\mu }_L$ and ${\lambda }_L$, respectively, which gave

$$B=\frac{4{\mu }_L\left({\mu }_L+{\lambda }_L\right)}{2{\mu }_L+{\lambda }_L}=\frac{E}{1-{\nu }^2}.$$

In this case, the dominant wavelength of the instability was given by

$${\lambda }_{c{r}_v}=2\pi h{\left(\frac{B\left(p+r_f\right)}{12G_{f}p}\right)}^{1/3}$$

and critical in-plane load due to growth was $N_{g_{cr}}=3\left(\frac{G_{f}p}{p+r_f}\right){\left(\frac{B\left(p+r_f\right)}{12G_{f}p}\right)}^{1/3}.$ Reorganizing these expressions yields

$${\lambda }_{c{r}_v}=\pi h{\left(\frac{2B}{3}\right)}^{1/3}{\left(\frac{\left(p+r_f\right)}{12G_{f}p}\right)}^{1/3}$$

and

$${\left(N_{g_{cr}}\right)}^{-1/2}{\left(\frac{2}{3}\right)}^{1/3}{B}^{1/6}={\left(\frac{\left(p+r_f\right)}{12G_{f}p}\right)}^{1/3}.$$

Substituting for ${\left(\frac{\left(p+r_f\right)}{12G_{f}p}\right)}^{1/3}$, we can remove $p$ from our expression for the dominant wavelength and show that

$${\lambda }_{c{r}_v}=\pi h\sqrt{\frac{B}{N_{g_{cr}}}}.$$

This equation indicates that, in the case of a viscoelastic foundation, the dominant wavelength is independent of the material properties of the foundation. Rather, it depends on the in-plane load $N_{g_{cr}}$, which is a linear function of ${\mathit{ϵ}}_g$.

Model of Time History of Epithelial Folding.

The onset of this growth-induced instability can be initiated by tiny irregularities in the initial deflection of the epithelial layer. If this “noise” is decomposed into Fourier components, those components with wavelengths that correspond to the dominant wavelength of the instability will grow the fastest and become visible. (To the best of our knowledge, this basic idea was first proposed by Biot (47) to explain the onset of folding instabilities during tectonic mechanics.)

Following Biot et al. (48), we simulated this initial noise $v_0\left(x\right)$ by creating $N$ peaks at random (and overlapping) locations along the layer, with amplitudes randomly selected between $-{\widehat{v}}_0$ and ${\widehat{v}}_0$. The resulting noise in the deflection was represented using the Fourier integral

$$v_0\left(x\right)=\underset{0}{\overset{\infty }{{\displaystyle \int }}}{e}^{-\gamma a}\left({\displaystyle \sum }_{i=1}^N{\varphi }_i{\widehat{v}}_0\cos \left[\gamma (x-{\widehat{x}}_i)\right]\right)d\gamma ,$$

where $N$ represents the number of peaks, $a$ is a measure of peak width, ${\widehat{x}}_i$ is the randomly generated location for an individual peak (between the values of $0$ and $L$, the length of the layer), ${\widehat{v}}_0$ is the amplitude of the fluctuations, and ${\varphi }_i$ is a randomly selected variable between −1 and 1.

From these initial conditions, the amplitude of the deflection $v(x,t)$ increased exponentially in time by $v\left(x,t\right)=v_0{e}^{kt}$ where the exponent $k$ was dependent upon how closely the wavelength of each Fourier component corresponded to the dominant wavelength of the instability. For the case of a growing elastic layer on a Maxwell-type foundation, for slow deformations, we get

$$\hspace{0.17em}k=\frac{r_f}{2G_f}\left(N_{g}h\gamma -\frac{B{\left(h\gamma \right)}^3}{12}\right),$$

where $h$ represents the thickness of the layer, $\gamma$ is related to the wavelength $\lambda$ of each Fourier component via $\gamma =2\pi /\lambda$, and (as above) $N_g=bEh{\mathit{ϵ}}_g$.

Discussion

Biological patterns are thought to arise primarily as a consequence of biochemical signaling. In several species, morphogens interact via reaction-diffusion kinetics to generate discrete patterns of gene expression in the embryo (14). Other (nonchemically mediated) mechanisms of pattern formation have received considerably less attention (30). Here, in the absence of any biochemical template, we have shown that purely physical interactions can drive biological patterning during branching morphogenesis of the embryonic airway epithelium. We took advantage of the mesenchyme-free culture system, which abolishes the prepattern of mesenchymal growth factors that are thought to coincide with locations of epithelial branches in the developing lung. Our data suggest that branches arise from a physical instability between the epithelium and its surrounding microenvironment. This folding is mediated by nonspatially patterned proliferation in the epithelium. Clear spatial patterns do not arise until branches have already formed, but it is unclear how the populations of highly proliferative cells, which are located at the tips of incipient branches, are either specified or maintained. The mechanical stresses induced by the initial folding of the epithelium may contribute to this spatial pattern, because mechanical stresses have been shown to control cell proliferation in 2D microfabricated tissues (31).

Importantly, mesenchyme-free branching is not unique to the developing lung. In the absence of mesenchyme, epithelial tissues from a variety of developing organs, including the salivary gland (32), kidney (33), and lacrimal gland (34), all generate similar branched structures when embedded in Matrigel. Even isolated fragments of intestinal epithelium produce similar geometries (35). Remarkably, in each case, proliferating cells are limited to the tips of the extending branches. Given the different tissue types, one would expect strikingly different biochemical microenvironments during morphogenesis, yet the topologies of the branched architectures are similar. It is possible that a conserved physical mechanism underlies the folding of each of these epithelial tissues.

It is unclear whether a similar instability drives epithelial branching in the intact lung. Smooth muscle differentiation has been shown to increase the stiffness of mesenchymal tissues (18), so it is possible that the spatial patterns of airway smooth muscle might constrain a growth-induced instability within the epithelium in vivo. Alternatively, localized differences in proliferation between the epithelium and smooth muscle might pattern epithelial buckling. In both cases, contractile forces generated by mesenchymal cells might also influence the mechanics of epithelial branching (36, 37).

Regardless, our data diverge from the generally accepted view that airway branching is driven by differential epithelial proliferation. Here, we found the opposite. Patterned proliferation does not induce branching, but is a consequence of the epithelial folding that takes place as new branches form. In a sense, proliferation is “downstream” of branch initiation (22). Together, these data suggest that physical mechanisms are sufficient to pattern the morphogenesis of developing epithelia—a result that strongly asserts that material properties and mechanical cues, in addition to the well-recognized role of biochemical signals, need to be considered in studies of tissue development (38–40).

It remains unclear exactly how epithelial cells might respond to physical cues in their surrounding microenvironment. Here, we show that cells in the developing airway epithelium are insensitive to changes in the elastic properties of the surrounding matrix. They may, however, be responsive to other physical properties in the microenvironment (e.g., plasticity, material inhomogeneity, etc.). Additional work using sophisticated cogels of Matrigel and other synthetic or naturally derived matrices, with tunable mechanical properties (41), will be necessary to more fully elucidate the mechanoregulatory mechanisms at work during epithelial morphogenesis. Ultimately, these problems will demand interdisciplinary collaborations between engineers, physicists, materials scientists, and cell and developmental biologists and will require new experimental (42) and computational (43) approaches to quantify the mechanical forces and material properties that shape developing tissues.

Methods

Mesenchyme-Free Culture.

Embryos were harvested from timed-pregnant CD1 (Charles River Laboratories) or vim-GFP (Mutant Mouse Regional Resource Center) mice at embryonic day 12–13, following protocols approved by the Princeton University Institutional Animal Care and Use Committee. Whole lung explants were dissected in ice-cold PBS using fine forceps. To isolate the airway epithelium, explants were incubated in 10 U/mL dispase (Invitrogen) on ice for 15 min; mesenchyme was then removed using fine tungsten needles (44). Isolated epithelial explants were embedded in undiluted growth factor-reduced Matrigel (BD Biosciences), Matrigel diluted in culture medium (DMEM/F12 supplemented with 0.5% FBS, penicillin, and streptomycin), or in gels consisting of 50% Matrigel and 0.5% methylcellulose (Sigma; average molecular weight ∼40,000). Gels were incubated at 37 °C for 30–60 min to set. Culture medium supplemented with FGF1 was then added, and explants were cultured for 48–72 h. Time-lapse movies were collected using a Nikon Ti-U inverted microscope customized with a spinning disk (BD Biosciences).

Quantifying Cell Proliferation.

Proliferating cells were detected using the Click-iT EdU Imaging Kit (Invitrogen). Briefly, mesenchyme-free explants were pulsed with 5-ethynyl-2′-deoxyuridine (EdU) for 1 h, then fixed with 4% (wt/vol) paraformaldehyde in PBS and washed with 0.3% Triton-X-100 in PBS. Fixed explants were then incubated with primary antibody against E-cadherin and, subsequently, AlexaFluor-conjugated secondary antibodies to simultaneously label the airway epithelium. Explants were then removed from the surrounding gel, and confocal stacks were captured. Three-dimensional reconstructions of the airway epithelium and proliferating cells were generated using Imaris (Bitplane). From the confocal stacks of E-cadherin, we used Amira (FEI Visualization Sciences Group) to create surface reconstructions of the airway epithelium. The surface mean curvature $\kappa$ was computed by $\kappa =\left({\kappa }_1+{\kappa }_2\right)/2$, where ${\kappa }_1$ and ${\kappa }_2$ are the principal curvatures. Cell proliferation was then mapped onto this reconstructed surface. Explants costained for EdU and Hoechst 33258 were used to quantify epithelial proliferation frequencies (Figs. S4 and S6).

Mechanical Testing.

We performed unconfined compression tests of cylindrical gel specimens to estimate the mechanical properties of Matrigel (Figs. S3 and S5). Before gelation, fluorescent microspheres (500 nm in diameter) were suspended in the liquid Matrigel at 4 °C. This solution was injected into a cylindrical PDMS mold of defined diameter, which was conformally sealed to the underlying tissue culture polystyrene, and incubated at 37 °C for 30 min. The PDMS mold was removed, and confocal stacks of the cylindrical gel were captured before loading. A 12-mm-diameter glass coverslip of known weight was then used to mechanically compress the gel, and a subsequent confocal stack of the compressed configuration was captured within 1–2 min of loading. As a first approximation, the observed change in thickness was used to estimate the (linear) elastic modulus of the gel.

Theoretical Model.

A detailed description of the model can be found in SI Text. The time history of folding simulations was solved using MATLAB.