Feedback-controlled epithelial mechanics: emergent soft elasticity and active yielding

TL;DR

The paper addresses how tissue-scale nematic order emerges from cell-scale processes and governs rheology during morphogenesis by introducing a minimal 2D vertex model where active stresses couple to nematic alignment with local elastic stress. Through analytical mean-field work and simulations, it demonstrates an isotropic--nematic transition and identifies a progression of states—soft nematic solid, plastic nematic solid, and nematic gas—organized by the effective activity $\alpha\beta$ and target shape index $P_0$, including long-range flows and defect dynamics in the plastic regime. A detailed phase diagram reveals a distinct active melting pathway and a triple point, emphasizing that activity can drive solid-like to fluid-like behavior while maintaining internal elastic stresses, which differs from standard fluidization. The findings illuminate how mechanical feedback enables active tissue remodeling and morphogenesis, providing a framework that bridges solid and fluid rheologies in epithelia and suggesting experimental signatures for stress and rheology across scales.

Abstract

Biological tissues exhibit distinct mechanical and rheological behaviors during morphogenesis. While much is known about tissue phase transitions controlled by structural order and cell mechanics, key questions regarding how tissue-scale nematic order emerges from cell-scale processes and influences tissue rheology remain unclear. Here, we develop a minimal vertex model that incorporates a coupling between active forces generated by cytoskeletal fibers and their alignment with local elastic stress in solid epithelial tissues. We show that this feedback loop induces an isotropic--nematic transition, leading to an ordered solid state that exhibits soft elasticity. Further increasing activity drives collective self-yielding, leading to tissue flows that are correlated across the entire system. This remarkable state, that we dub plastic nematic solid, is uniquely suited to facilitate active tissue remodeling during morphogenesis. It fundamentally differs from the well-studied fluid regime where macroscopic elastic stresses vanish and the velocity correlation length remains finite, controlled by activity. Altogether, our results reveal a rich spectrum of tissue states jointly governed by activity and passive cell deformability, with important implications for understanding tissue mechanics and morphogenesis.

Figures (11)

• Figure 1: (a) Schematic of the active feedback loop between the active extensile stress ${\bm{\sigma }}^{\mathrm{act}}$ generated by cellular fibers and its alignment with the local elastic stress ${\bm{\sigma }}^{\mathrm{el}}$. Here $\alpha$ is the activity and $\beta$ is the alignment strength. Their product $\alpha\beta$ determines an effective activity that controls the phase behavior. (b) Phase diagram in terms of the effective activity $\alpha\beta$ and the target cell shape index $P_0$. (c) Representative snapshots with velocity fields (yellow arrows) corresponding to the parameter combinations labeled by black stars in the phase diagram [see Video S3 in Supplemental Material]. The color map is magnitude of the nematic order parameter.
• Figure 2: (a) Diagram of active force induced by extensile nematic stress. The green stick represents the nematic order ${\bf{Q}}_J$ of cell $J$ and the green arrows denote the extensile active forces ${\bf{F}}^{\rm{act}}$ induced by ${\bf{Q}}_J$. $\bf n_{e_1}$ is the unit vector normal to edge $e_1$ pointing from cell $J$ to cell $I$. (b) Schematic of the uniform deformation of hexagonal cells used for the mean-field analysis. (c,d) Mean-field values of the order parameter $q$ and the shape anisotropy $s$ versus $\alpha\beta$, for (c) $m\!=\!-1$ and (d) $m\!=\!1$.
• Figure 3: (a) Tissue snapshots at different activity for $m\!=\!-1$. The color is the magnitude of the nematic order parameter. (b) Left: order parameter $q$ (green circles) and mean shape anisotropy $s$ (orange squares); Right: mean cell velocity $\langle v \rangle$ (blue circles) and T1 transition rate $k_{\rm T1}$ (magenta squares) as functions of $\alpha\beta$. All results are for $P_0=-0.5$.
• Figure 4: (a) Snapshots of the cell shape field (left, red lines) and the velocity field (right, blue arrows) at $t\!=\!400$ and $t\!=\!1.6\times10^4$ in the plastic solid regime ($\alpha\beta\!=\!2.4$). (b) Temporal evolution of mean cell velocity $\langle v \rangle$ and the number of $\pm 1/2$ defect pairs $N_{\rm{dp}}$ in isotropic, soft nematic, and plastic nematic solids, corresponding to $\alpha\beta=0.8,\ 1.6,\ 2.4$, respectively. In the isotropic solid ($\alpha\beta=0.8$), both velocity and number of defect pairs are essentially zero at all times and the curves overlap. (c) Evolution of $N_{\rm{dp}}$ over time in the soft nematic solid regime ($\alpha\beta\!=\!2$) for different cell numbers $N$. (d) Snapshots of the cell shape (white lines) and order parameter (color bar) for $N = 10000$ cells, where two defect pairs exist. All results are for $P_0=-0.5$.
• Figure 5: (a) Shear stress–strain curves for varying $\alpha\beta$ showing the mean deviatoric stress $\langle\lvert\tilde{\boldsymbol{\sigma}}\rvert\rangle \! =\!{\langle\lvert\boldsymbol{\sigma}_{\!J}\!-\!\tfrac{1}{2} (\mathrm{tr}\,{\boldsymbol{\sigma }}_{\!J})\mathbf{I}\rvert\rangle}_{\!J}$ as a function of strain $\gamma$. The initial shear modulus $G_0$ is obtained from the slope under an infinitesimal strain, as highlighted by the dashed black line. $\gamma_c$ is the critical strain at which rigidity emerges, as shown by the arrows along the strain axis. $\tilde{\sigma}_{\!\mathrm{max}}$ and $\tilde{\sigma}_{\!\mathrm{py}}$ denote the peak and post-yield shear stress, respectively. (b) Evolution of $G_0$ (dashed black line), $\gamma_c$ (solid black line), $\tilde{\sigma}_{\!\mathrm{max}}$ (dotted green line), and $\tilde{\sigma}_{\!\mathrm{py}}$ (dash-dotted orange line) versus $\alpha\beta$. (c--e) Representative snapshots of (c) isotropic ($\alpha\beta\!=\!0.6$), (d) soft nematic ($\alpha\beta\!=\!1.4$), and (e) plastic nematic ($\alpha\beta\!=\!2.2$) solid tissues under shear deformation. The corresponding snapshots of deviatoric stress are shown in Fig. \ref{['FigureS3']}. All results are for $P_0=-0.5$.