Theory of epithelial elasticity

We propose an elastic theory of epithelial monolayers based on a two-dimensional discrete model of dropletlike cells characterized by differential surface tensions of their apical, basal, and lateral sides. We show that the effective tissue bending modulus depends on the apicobasal differential tension and changes sign at the transition from the flat to the fold morphology. We discuss three mechanisms that stabilize the finite-wavelength fold structures: Physical constraint on cell geometry, hard-core interaction between non-neighboring cells, and bending elasticity of the basement membrane. We show that the thickness of the monolayer changes along the waveform and thus needs to be considered as a variable rather than a parameter. Next we show that the coupling between the curvature and the thickness is governed by the apicobasal polarity and that the amplitude of thickness modulation along the waveform is proportional to the apicobasal differential tension. This suggests that intracellular stresses can be measured indirectly by observing easily measurable morphometric parameters. We also study the mechanics of three-dimensional structures with cylindrical symmetry.

Figure 1

Schematic of the model epithelial cell characterized by different surface tensions on apical, basal, and lateral sides [(a), left]. The right panel shows a bent segment of tissue parametrized by midline curvature $C$, epithelial thickness $L$, and depth $r$. In the discrete representation this body can be treated as the model cell. Schematic cross section of three epithelial waveforms in the discrete representation (b), parametrized by cell side lengths ($L_a$, $L_b$, and $L_l$) and the continuous waveform, parametrized by the midline local curvature $C$ and local tissue thickness $L$. Gray shading denotes the transition from the discrete to the continuum description.

Figure 2

Parametrization of tissue cross-section by the angle $\psi \left(s\right)$ and tissue thickness $l\left(s\right)$ (a). In a single waveform of wavelength $\lambda$, the arc length $s$ goes from $s=0$ to $s=S$. A comparison between minimal-energy shapes obtained by the discrete model [17] and minimal-energy shapes computed by minimizing the elastic functional [Eq. (2)] at $\beta =2.6$ for $\alpha =4.2,5.0,5.8,\text{and}6.6$ (b). Continuum-theory waveforms are computed at the same wavelength and number of cells per waveform as in the discrete model. Minimal-energy shapes of a segment of epithelium at $\alpha =\beta =1.5$ for different values of compressive force $\gamma =-0.036,-0.041,-0.048,\text{and}-0.052$; $\gamma$ is defined in Sec. 3a (c). Equilibrium shapes of a piece of paper attached to a flat surface, the opposite ends being fixed at the distances shorter than the length of the paper (d). Thick cyan (light gray) line is added to highlight the contour. The equilibrium states of the folds correspond to the same family of curves as that of the tissue midline in panel (c).

Figure 3

Force exerted on a segment of tissue with cross-section area $A=25$ and $\alpha +\beta =6.8$, fixed at $\lambda =6$, as a function of apicobasal differential tension $\alpha -\beta$ [red (dark gray) curve]. Also included are the bending modulus $k_{\mathrm{e}}$ [green (light gray) curve] and the amplitude of tissue thickness modulation $\delta l/l_0$ (black line). At $\left|\alpha -\beta \right|=\sqrt{2}$ both $\gamma$ and $k_e$ change sign and for $\left|\alpha -\beta \right|>\sqrt{2}$ (shaded region) fold morphologies are energetically favorable over the flat state. Insets are equilibrium shapes at fixed $\lambda =6$ and $\alpha +\beta =6.8$ for $\alpha -\beta =-2.8,-\sqrt{2},0,\sqrt{2}$, and 2.8 (left to right).

Figure 4

Minimal-energy shape of the tissue cross section for $\alpha =4.2$ and $\beta =2.6$ obtained by the discrete model (a). In this case the groove is energetically favorable over the crest and contains tall triangular cells, whereas the crest consists of trapezoidal cells. The product of curvature and thickness $cl$ along the waveform shown in panel (b) is bounded between $-2$ and 2 which are the values corresponding to basally constricted cell and apically constricted cell, respectively. Minimal-energy shape of the tissue cross section for $\alpha =6.6$ and $\beta =2.6$ obtained by the continuum version of our model (c). The apical side of the groove makes self-intersection loops in absence of the physical constraint of positive cell side lengths. Minimal-energy shape of the tissue cross section for $\alpha =5$ and $\beta =2.6$ obtained by the discrete model (d). Steric repulsion acts at points where non-neighboring cells touch [red (dark gray) arrows].

Figure 5

Schematic of the model waveform (a). The groove is characterized by a negative curvature $-c$ and tall triangular cells, whereas the crest has opposite curvature $c$ and consists of trapezoidal cells. Curvature of the basement membrane is larger in the crest, where cells' basal sides are shorter than their apical sides. Energy terms $w_e$, $w_{\mathrm{bm}}$, and their sum $w_e+w_{\mathrm{bm}}$ at $\alpha +\beta =6$, $\alpha -\beta =-3$, and $\kappa =0.1$ (b). Equilibrium wavelength as a function of $\kappa$ at $\alpha +\beta =4$ for different values of differential tension $\left|\alpha -\beta \right|=1.5,2$, and $2.5$ (c). Red (dark gray) curves correspond to the $\alpha <\beta$ case, whereas green (light gray) curves correspond to the $\alpha >\beta$ case.