Vertex Model Mechanics Explain the Emergence of Centroidal Voronoi Tiling in Epithelia

TL;DR

Problem addressed: why do epithelial tilings resemble centroidal Voronoi tessellations and how is this organization controlled by mechanics? Approach: combine vertex-model (VM) simulations with a mean-field analytical mapping to the quantizer energy $E_q$, establishing $E_q \propto E_{\rm VM}$ in the solid phase and showing CVT configurations minimize VM energy. Key findings: CVT-like order emerges near the solid regime (with $p_c \approx 3.81$ and genome honeycomb at $p_0 \le 3.72$); isotropic/oscillatory stretch lowers the effective shape index $p_0^{\mathrm{eff}}$, driving CVT-like configurations and producing measurable signatures such as increased variance in $p$ and reduced hexagon fraction $P(6)$. Significance: provides a geometric framework to infer tissue stresses from morphology and to diagnose rigidity, remodeling, and stretch in living epithelia, with implications for developmental mechanobiology.

Abstract

Epithelia are confluent cell layers that self-organize into polygonal networks whose geometry encodes their mechanical state. A principal driver is the tunable contractility of the actomyosin cortex, which links cell-junction tension to tissue architecture. Notably, epithelial tilings frequently resemble centroidal Voronoi tessellations (CVTs), yet the physical origin of this resemblance has remained unclear. Here, using a minimal vertex model that relates cell shape to a mechanical energy, we show that CVT-like patterns arise naturally in the solid (rigid) regime of tissues. Analytical theory reveals that isotropic strain minimization drives cell centroids toward Voronoi configurations, a result we corroborate with a analytical mean-field formulation of the vertex model. We further demonstrate that physiologically relevant perturbations -- such as cyclic stretch -- shift tissues into distinct, geometrically disordered CVT states, and that these shifts provide quantitative, image-based readouts of mechanical state. Together, our results identify a mechanical origin for CVT-like organization in epithelia and establish a geometric framework that infers tissue stresses directly from morphology, offering broadly applicable metrics for assessing rigidity and remodeling in living tissues.

Figures (3)

• Figure 1: Vertex models (VM) approach CVTs as the shape index $p_0$ decreases. The VM--CVT deviation metric $\triangle_{\mathrm{CVT}}$, which measures the average dimensionless difference between a VM tessellation and its corresponding centroid-seeded Voronoi diagram, decreases monotonically with the shape index $p_0$. Because lowering $p_0$ generally increases the mechanical stiffness of VM configurations, stiffer states exhibit patterns increasingly similar to centroidal Voronoi tessellations (CVTs). Representative VM networks (black) and their closest CVTs (red) at selected values of $p_0$ are shown in the top-row insets. Cells in the solid phase---i.e., for $p_0$ below the rigidity transition at $p_0 \simeq 3.81$ (blue dashed line)---produce near-perfect CVT-like patterns. Further decreasing $p_0$ deepens the solid regime and yields increasingly isotropic arrangements, with $p_0 = 3.72$ producing a nearly ideal honeycomb lattice. The typical rigidity transition at $p_0 = 3.81$ is indicated by the blue dotted line; notably, $\triangle_{\mathrm{CVT}}$ varies smoothly across this transition.
• Figure 2: Oscillatory stretching drives fluid cells toward CVT patterns by lowering the average shape index $\langle p\rangle$.(a) The VM--CVT deviation $\triangle_{\mathrm{CVT}}$ as a function of $p_0$ under varying fractional stretch $e_{\mathrm{eff}}$. Increasing stretch systematically reduces $\triangle_{\mathrm{CVT}}$ at fixed $p_0$. For oscillatory stretching with maximum amplitude $e$, the effective stretch is taken as the time-averaged value $e_{\mathrm{eff}} = e / 2$. (b) Replotting all simulation data for varying $p_0$ and $e$ (Fig. 2a) in terms of the mean cell shape $\langle p \rangle$, with colors indicating the corresponding $p_0$. The collapse of these curves onto the unstretched reference curve (black) indicates that stretch drives cells toward CVT-like configurations by effectively reducing the target shape index $p_{0}^{\mathrm{eff}}$. (c) Replotting all simulation data directly in terms of $p_0^{eff}=\frac{p_0}{1+e}$ we show a good curve collapse in the fluid phase, showing that the reduction of $\triangle_{\rm CVT}$ is directly due to the stretch-induced rescaling of $p_0^{eff}$.
• Figure 3: Distinguishing unstretched and stretched CVT-like states at constant $\langle p \rangle$ via persistent structural disorder. Representative unstretched (a) and stretched (b) vertex-model tissues at fixed $\langle p \rangle = 3.83$ are shown, with the VM tessellation (left) in black and the corresponding cell shape indices color-coded (right). (c) Stretched tissues exhibit a systematically larger variance in $\langle p \rangle$ than unstretched tissues, across all simulations with $3.7 < p_0 < 4.5$ and $0 < e < 0.5$.(d) At fixed $\langle p \rangle$, the proportion of hexagons $P(6)$ decreases with increasing stretch, reflecting persistent structural disorder inherited from the fluid-like initial state.Although unstretched and stretched tissues appear similar when compared using average observables (Fig. \ref{['fig2']}b), they are clearly distinguishable through their cell-to-cell variability (c) and monolayer polygonal structure (d). Because both $P(6)$ and $\langle p \rangle$ are purely observable quantities, they provide an experimentally accessible readout of the underlying applied stretch.