Thickness of epithelia on wavy substrates: measurements and continuous models

TL;DR

This study quantifies how substrate curvature affects epithelial thickness by culturing MDCK epithelia on sinusoidally corrugated substrates and measuring valley–ridge thickness differences. It tests three continuum descriptions—thin liquid film, thin elastic film, and thin elastic film with apical active stress—against the data, finding that surface tension and elasticity alone yield qualitative agreement but fail to match the measurements quantitatively. Incorporating an apical active stress substantially improves agreement, with fitted parameters suggesting differing apical contractility between WT and E-cadherin–GFP–expressing epithelia. The work links cellular-level active stresses to tissue-scale curvature responses, providing a framework to study curvature-driven morphogenesis in epithelia.

Abstract

We measured the thickness of MDCK epithelia grown on substrates with a sinusoidal profile. We show that while at long wavelength the profile of the epithelium follows that of the substrate, at short wavelengths cells are thicker in valleys than on ridges. This is reminiscent of the so-called « healing length » in the case of a thin liquid film wetting a rough solid substrate. We explore the ability of continuum mechanics models to account for these observations. Modeling the epithelium as a thin liquid film, with surface tension, does not fully account for the measurements. Neither does modeling the epithelium as a thin incompressible elastic film. On the contrary, the addition of an apical active stress gives satisfactory agreement with measurements, with one fitting parameter, the ratio between the active stress and the elastic modulus.

Figures (6)

• Figure 1: MDCK epithelia grown on PDMS substrates with sinusoidal profiles. The nuclei appear in blue, F-actin in grey and the apical membranes in red. (A) Top view and (B) cross-section of MDCK-EcadGFP cells, E-cadherin appears in green and both apical membranes and fibronectin appear in red. (C) Cross-section view of MDCK-WT cells, fibronectin appears in green. Scale bars = 20 $\mu m$.
• Figure 2: Measured difference in the thickness of the epithelial cells in the valleys and on the ridges of the sinusoidal substrates as a function of the inverse of its wavelength. Error bars show the 95% confidence intervals.
• Figure 3: Experimental values of the difference $\Delta H$ in the thickness of the epithelial cells in the valleys and on the ridges of the substrates, divided by the amplitude $A$ of the sinusoidal profile, and their best fits with Eq. \ref{['eq_delta_H_vs_lambda']}. Error bars show the 95% confidence intervals.
• Figure 4: Experimental values of the dimensionless thickness difference $\Delta H / A$, as a function of the dimensionless substrate wave vector $2\pi H_0/\lambda$, and plots of the model, $\left[1-F\left(\frac{2\pi H_0}{\lambda},\frac{\pi \gamma}{\mu \lambda}\right)\right]$ (cf. Eq. \ref{['elastic_model']}). The gray area corresponds to the interval $\pi \gamma/\mu \lambda = 0-0.08$ which is the largest interval that can be reached assuming the smallest value of $\lambda$ (40$\mu$m) and the highest bound for $\gamma/\mu$ of $1\mu$m.
• Figure 5: Elastic model with active stress and surface tension for epithelia. (A) Reference state: a soft flat epithelium sits on top of a rigid substrate. (B) Final shape of the epithelium when the substrate is deformed into a sinusoidal profile. Panels (C) to (F): respective contribution of stiffness, active surface stress, apical tension and epithelium thickness to the profile of the epithelium on sinusoidal substrate. (C) Log-linear plot of the dimensionless thickness difference $\Delta H/A$ as a function of the epithelium stiffness $\mu$ in kPa. The thickness $H_0$ of the epithelium is set at 10 $\mu$m, its apical tension $\gamma$ at 5mN/m, the active stress $\sigma^{act}$ at 0 and the wavelength $\lambda$ of the sinusoidal substrate at 50 $\mu$m. The insets show the limits for a very soft and a very stiff epithelium. (D) Effect of $\sigma^{act}$ on $\Delta H/A$; $H_0$=10 $\mu$m, $\gamma$=5mN/m, $\mu$=1kPa, and $\lambda$=50 $\mu$m. (E) Effect of $\gamma$ (in mN/m) on $\Delta H/A$; $H_0$=10 $\mu$m, $\mu$=1kPa, $\sigma^{act}$=0 and $\lambda$=50 $\mu$m. (F) Effect of $H_0$ (in $\mu$m) on $\Delta H/A$; $\gamma$=5 mN/m, $\mu$=1kPa, $\sigma^{act}$=0 and $\lambda$=50 $\mu$m.