Solid-to-solid transition in dense assemblies of elongated cells

Technical Summary: Solid-to-Solid Transition in Dense Assemblies of Elongated Cells

Problem Statement
In confluent biological tissues, cell morphology ranges from isotropic epithelial shapes to highly elongated endothelial forms. Standard vertex models describe tissue mechanics by controlling rigidity via a "target shape index" ($p_0$), defined as the ratio of a cell's preferred perimeter to the square root of its preferred area. In these standard models, increasing $p_0$ lowers energy barriers for neighbor exchanges, eventually driving a solid-to-fluid transition where tissue rigidity vanishes. However, many biological cell types (e.g., myoblasts, neural progenitors, endothelial cells) possess an intrinsic, passive preference for elongated morphologies driven by cytoskeletal organization or polarity, rather than an isotropic area-perimeter competition. The central question addressed is whether this autonomous cell elongation fluidizes the tissue as predicted by shape-index arguments, or if it establishes a distinct mechanical route to rigidity.

Methodology
The authors extend the standard vertex model of confluent cell sheets by introducing a "Cell Elongation Elasticity" (CEE) term.

• Model Framework: The tissue is modeled as a polygonal tiling where degrees of freedom are vertex positions $\mathbf{r}_i$. The total mechanical energy $E$ is the sum of the standard vertex model energy ($E_{svm}$) and the CEE energy ($E_{cee}$).
• Energy Terms:
  • $E_{svm}$ penalizes deviations from a target area $A_0$ and perimeter $P_0$.
  • $E_{cee} = \sum_{\alpha} \frac{1}{2}K_Q [\text{tr}(\mathbf{Q}_\alpha^2) - \frac{1}{2}q_0^2]^2$ penalizes deviations from a preferred anisotropic shape $q_0$. Here, $K_Q$ is the elongation stiffness, and $\mathbf{Q}_\alpha$ is a second-order tensor describing cell-shape anisotropy based on cell edges.
• Dynamics and Simulation: The system evolves via overdamped dynamics balancing friction against shape-relaxation forces derived from the total energy. Simulations initialize with $N_c = 10^3$ cells on a regular hexagonal lattice (and tested on random Voronoi lattices) with periodic boundary conditions. Parameters are rescaled by $\sqrt{A_0}$ and stress scales.
• Analysis: The authors derive analytical expressions for mechanical energy, shear modulus ($G_{xy}$), and yield stress ($\sigma_{yield}$) under affine transformations. They compare these theoretical predictions with numerical simulations across varying $K_Q$ and $P_0$.

Key Results

1. Solid-to-Solid Transition: Contrary to the standard expectation that elongation fluidizes tissue, the introduction of CEE drives a solid-to-solid transition. As the elongation stiffness $K_Q$ increases, the tissue transitions from an ordered, isotropic solid state to a disordered, anisotropic solid state.
2. Persistence of Rigidity: In the anisotropic solid state (high $K_Q$), the tissue retains finite shear rigidity and a finite yield stress. The long-time shear modulus $G_{xy}^{(LT)}$ decreases to zero at a critical stiffness but then rises again to a plateau, indicating the restoration of stiffness.
3. Mechanism of Stiffening:
   • For small $K_Q$ (isotropic regime), CEE contributes negatively to the shear modulus.
   • For large $K_Q$ (anisotropic regime), cell elongation induces increased intercellular tension ($T_{\alpha\beta}$). This tension leads to a linear strain-stiffening regime governed by the cell perimeter stiffness $K_P$, restoring positive, finite shear modulus.
4. Yield Stress Behavior: The yield stress remains finite even beyond the point where the short-time shear modulus vanishes. The system exhibits a transition from brittle yielding (abrupt stress drop) to ductile yielding (smoother transition) as $K_Q$ increases further.
5. Phase Diagrams:
   • For $P_0 < P_0^*$ (standard solid regime), increasing $K_Q$ triggers the solid-to-solid transition.
   • For $P_0 > P_0^*$ (standard fluid regime), even an infinitesimal $K_Q$ is sufficient to solidify the tissue, shifting the critical point to $K_Q \to 0^+$.
6. Robustness: These findings hold for both regular hexagonal and random Voronoi initial patterns, and are robust to variations in the preferred elongation $q_0$.

Significance and Claims
The paper claims to decouple cell shape from tissue rheology, challenging the inference that elongated cell morphologies necessarily imply fluid-like mechanics.

• Novel Control Parameter: The authors establish CEE as an independent control parameter for tissue mechanics, distinct from the target shape index.
• Microscopic Mechanism: The work supplies a microscopic mechanism for solid-to-solid transitions in biological tissues, linking how cells autonomously set their shape to how the tissue resists deformation.
• Contrast with Active Models: Unlike active polar or nematic models where elongation triggers spontaneous flows and fluidization, this passive mechanical equilibrium model demonstrates that autonomous elongation can maintain or restore tissue rigidity.
• Implications for Morphogenesis: The results suggest that tissues composed of elongated cells (e.g., C2C12 myoblasts or HUVECs) may possess significant mechanical resistance, which must be accounted for when modeling morphogenetic processes like embryo folding or wound closure.

The authors conclude that measuring the stiffness of strongly elongated cell cultures would directly test the link between CEE and tissue rheology, cautioning against relying solely on cell morphology to infer tissue fluidity.