Shaping epithelial lumina under pressure

Abstract

The formation of fluid- or gas-filled lumina surrounded by epithelial cells pervades development and disease. We review the balance between lumen pressure and mechanical forces from the surrounding cells that governs lumen formation. We illustrate the mechanical side of this balance in several examples of increasing complexity, and discuss how recent work is beginning to elucidate how nonlinear and active mechanics and anisotropic biomechanical structures must conspire to overcome the isotropy of pressure to form complex, non-spherical lumina.

Keywords: biophysics; cell mechanics; epithelial cells; lumen formation.

Figure 1.. Examples of pressurised epithelial lumina.

(A) MDCK cysts have a typical epithelial cell polarisation, with apical surfaces facing the lumen and basal surfaces opposite, facing the surrounding matrix, and lateral surfaces at cell–cell interfaces. Tight junctions separate the apical and lateral surfaces. (B) The lumina of MDCK cysts can take on a variety of shapes depending on culture conditions and genetic perturbations: Examples of a wild-type cyst and a Zonula occludens-1 and Zonula occludens-2 double knockout (ZO-KO) [16]. The wild-type cyst is spherical, while in the ZO-KO the apical cell surfaces bulge out toward the basal side. Scale bar: 10 μm. (C) Bile canaliculi lumina are formed by just two hepatocytes. The apical cortices of the two cells surround most of the lumen, with the junctional complex located at the cell–cell interface. (D) Growth of bile canaliculi: Nascent bile canaliculi are spherical lumina that elongate to form the bile canaliculi network [29]. Striations across the tubular bile canalculi are ‘apical bulkheads’. Scale bar: 10 μm. (A) and (B) have been modified from [16] under a CC-BY-NC license, and (D) has been modified from [29] under a CC-BY license.

Figure 2.. Mechanics of spherical lumina.

(A) The Young–Laplace law relates the pressure $p$ inflating a spherical surface of radius $r$ to the tension $t$ in the surface. (B) Plot of the pressure–radius relationship predicted by the Young–Laplace law at constant tension $t$: If a lumen of pressure $p_1$ has radius $r_1$, then a lumen of larger pressure $p_2>p_1$ has smaller radius $r_2<r_1$. (C) Elastic model of lumen inflation: A spherical lumen of undeformed radius $r_0$ inflates to one of radius $r$ under an imposed pressure $p$. (D) The linear elastic energy $W$ of the system has a minimum at $r=r_0$, while the radius of the lumen is set by the minimum (highlighted point) of the enthalpy $G$, which is given in eqn (3). This minimum exists for $p<p_{\ast }$. (E) The resulting pressure–radius relation is given by eqn (4), and has a maximum at $p_{\ast }$; for $p>p_{\ast }$, the lumen radius diverges, $r\to \mathrm{\infty }$. This divergence is replaced by an inflation jump of finite size if the linear elastic energy density is replaced with a nonlinear Mooney–Rivlin density [42, 43]. (F) Mean-field model of a spherical cyst of inner radius $r$ and outer radius $R$, consisting of $N$ identical cells which are incompressible square frusta of inner side $x$, outer side $X$, and height $R-r$. Their apical area $A_{\text{a}}$, basal area $A_{\text{b}}$, lateral area $A_{\ell }$, and apical perimeter $P_{\text{a}}$ define the mechanical energy $W$ of the cyst. (G) The pressure–radius relation that results from this model in the limit of a large cyst of many cells, expressed by eqn (7), is qualitatively similar to that of the elastic model, but differs at a quantitative level, because of the emergent nature of the mechanics at the cyst scale.

Figure 3.. Mechanics of non-spherical lumina: Spontaneous symmetry breaking due to mechanical nonlinearities.

(A) Spherical cyst of radius $R$ and small lumen radius $r\approx 0$, and elongated, tubular cyst consisting of two hemispherical caps of radius $R$ and a cylindrical middle region of radius $R$ and length $L$. The $N$ cells in the cyst are either square pyramids of side $R$ or cylindrical sectors of radius $R$, thickness $\ell$ and length $X$. A fraction $f$ of the cells are in the cylindrical region. (B) Plot of the difference $G_{\text{tube}}-G_{\text{sphere}}$ of the enthalpies of the tubular and spherical cysts, showing a negative minimum value at some $f>0$, highlighted by the large mark, and illustrating spontaneous symmetry breaking.