Figure 1

The structure of simple epithelium (a) consisting of a single layer of elongated prismatic cells (SEM of mucosal epithelium in rat small intestine; image courtesy of Dr. Roger Wagner, University of Delaware) is uniquely described by their bases that form a polygonal tiling partly outlined at the apical surface. The shape of cells (b) is a prism with a base of area $A$ and perimeter $L$. In our model, $A$ and $L$ are assumed to be fixed and identical in all cells described by base reduced area $a=4\pi A/L^2$.Reuse & Permissions

Figure 2

(Color online) Typical evolution of the equilibrium structure of the tiling during a simulation run: the initially regular tiling consisting exclusively of hexagons with identical reduced areas $a$ is equilibrated after a few 1000 Monte Carlo steps including a random edge length change or a T1 transformation. Upon equilibration, the frequency of hexagons decreases and the frequencies of the other polygon classes gradually increase. The data shown here correspond to an $a=0.70$ tiling (containing 400 polygons) whose equilibrium structure is disordered and includes four-, five-, six-, seven-, eight-, and nine-sided tiles.Reuse & Permissions

Figure 3

(Color online) Representative snapshots of equilibrium theoretical tilings at polygon reduced areas $a=0.74$, 0.78, 0.82, 0.86, and 0.90. For values of polygon reduced area $a$ smaller than 0.785 the tiling is disordered and includes four-, five-, six-, seven-, eight-, and nine-sided tiles whose frequencies barely depend on $a$; each polygon class is encoded by color or a shade of gray. Beyond $a=0.785$, the frequency of hexagons increases with $a$ at the expense of the nonhexagonal polygons. At $a=0.865$ the order-disorder transition takes place; for $a>0.865$ all polygons in the tiling are perfectly sixfold coordinated.Reuse & Permissions

Figure 4

(Color online) Frequencies of polygon classes as a function of reduced area: the order-disorder transition at $a=0.865$ is marked by a dramatic change in the frequencies of five-, six-, and seven-sided polygons, whereas the discontinuity of the frequencies of quadrilaterals, pentagons, hexagons, and heptagons at the onset of the robust disordered phase at $a=0.785$ is less pronounced.Reuse & Permissions

Figure 5

Theoretical phase diagram of proliferating adhering contractile cells (adapted from Fig. 1 in Ref. [11]): depending on the effective line tension $t$ and contractility $c$, the tissue can be either disordered or ordered. In the plane spanned by the normalized line tension $\overline{t}=t/\kappa {\left[A^{\left(0\right)}\right]}^{3/2}$ and contractility $\overline{c}=c/\kappa A^{\left(0\right)}$ (here $\kappa$ denotes the elastic coefficient and $A^{\left(0\right)}$ is the preferred cell area), the order-disorder transition is described by a linear function of negative slope (dashed line). For negative values of $\overline{t}$ shown here, this model is mechanically very similar to ours except for the cell division. Each of the superimposed solid straight lines through the origin connects points in the $\left(\overline{c},\overline{t}\right)$ plane that have identical values of the preferred cell reduced area $a=\text{const}$ and thus identical structure; the lines are labeled by the value of $a$ that they represent. The order-disorder transition takes place at a fixed value of $a\approx 0.89$ very close to our value of $a=0.865$ (which virtually overlaps with the dashed line).Reuse & Permissions

Figure 6

(Color online) Frequencies of polygon classes in the Drosophila embryo germband at stage 6 (a) and late stage 8 (b): experimental data (open columns, data from Ref. [13]) agree very well with our theoretical predictions for $a=0.853$ and $a=0.75$, respectively (solid columns). From the experimental data [Figs. 2b and 2c in Ref. [13]] we also extracted the distribution of polygon reduced area in the germband shown in panels (c) and (d). The two distributions peak at the average value of $⟨a⟩=0.87$ and $⟨a⟩=0.78$, respectively. Although somewhat larger, these values are quite consistent with the two reduced areas of our best-fit monodisperse tilings. The dispersion of the distribution $\left({\sigma }_a\right)$ at stage 8 is larger than at stage 6. The bin size used here is $\Delta a=0.33$.Reuse & Permissions

Figure 7

(Color online) Structure of Drosophila wing disk, Hydra tail epidermis, and Xenopus epidermis in terms of frequencies of polygon classes (data from Ref. [10]). The three sets of experimental data agree nicely with the theory proposed in Ref. [10]. However, they are equally well described by our $a=0.79$ tiling with the polygon class frequencies virtually identical to that of Ref. [10]. The $a=0.75$ tiling reproduces the observed frequency of quadrilaterals better.Reuse & Permissions

Figure 8

Statistical-mechanical properties of tilings. (a) Quadron PDF for representative polygon reduced areas corresponding to disordered tilings. The PDFs are increasingly more narrow and less right-skewed as $a$ is increased and agree with the gamma distribution (solid curves) except for the right tail, which is suppressed by the fixed-area constraint that each polygon must satisfy. The gamma distribution shape parameter $k$, which increases with polygon reduced area $a$, is an estimate of the number of quadrons in correlation clusters. (b) Quadrons, the independent degrees of freedom in a two-dimensional (2D) cellular network, are convex quadrilaterals spanned by a vertex, midpoints of adjacent edges, and polygon centroid. (c) The Gibbs entropy of the equilibrium tiling is a decreasing function of $a$ and bears no obvious signature of the onset of the robust disordered phase and the disordered-hexagonal transition, suggesting that they are continuous. Inset: at the beginning of the simulation run, the entropy increases until the initial hexagonal state reaches equilibrium after a few 1000 steps. Compared here are the entropies of an $a=0.81$ tiling in a box containing 400 and 1600 polygons (black and gray curve, respectively) to show that finite-size effects are negligible in a system of $\sim 100$ tiles.Reuse & Permissions