🔬 condensed matter

Discontinuous buckling from cytoplasmic mechanics in vertex models of epithelia

This paper demonstrates that incorporating cytoplasmic mechanics into vertex models of epithelia reveals a discontinuous buckling transition under compression, a qualitative change in tissue-scale behavior explained by tricritical points in a sixth-order Landau theory.

Original authors: Chandraniva Guha Ray, Pierre A. Haas

Published 2026-10-05

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CC BY 4.0

Original authors: Chandraniva Guha Ray, Pierre A. Haas

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Tissues in living animals are not just bags of cells; they are structured, mechanical materials that bend, fold, and hold their shape. To understand how these tissues behave, scientists often look at the cell surface, the outer boundary of each cell. This surface is reinforced by a contractile layer of proteins, acting like a tight skin, and is glued to neighboring cells and the surrounding environment. For years, models used to simulate how these tissues change shape have focused almost entirely on this surface tension and adhesion. They treat the inside of the cell, the cytoplasm, as a passive filler, ignoring the complex soup of proteins and structures suspended within it. However, this interior is not empty space; it is a dense, crowded environment that resists being squeezed. The question of whether this internal resistance changes how a whole tissue folds has remained largely unexplored until now.

A team of researchers at the Max Planck Institute for the Physics of Complex Systems and the Center for Systems Biology in Dresden has set out to fill this gap. They built a simplified computer model of a single layer of cells, similar to the lining of an organ, and added a specific mechanical cost to the energy required to squeeze the cell's interior. In their simulation, they imagined the cell's internal contents as a collection of polymers that become harder to confine as the cell gets smaller. By compressing this simulated layer of cells from the sides, they watched how the tissue responded. The results were surprising. While the tissue always buckled, or bent, under pressure, the way it buckled changed dramatically depending on the stiffness of the internal contents.

In the absence of this internal resistance, or when it was very weak, the tissue bent smoothly and gradually as the pressure increased. This is a continuous transition, where the curve grows slowly from flat to folded. However, the researchers found that when they increased the internal stiffness to a specific, intermediate range, the behavior changed completely. Instead of bending slowly, the tissue suddenly snapped into a folded shape. One moment it was flat, and the next, under a tiny bit more pressure, it jumped to a significantly bent state. This is a discontinuous transition, a sudden jump rather than a smooth curve. Even more unexpectedly, if they made the internal contents even stiffer, the tissue returned to bending smoothly again. The sudden snap only happened in that middle range of stiffness.

To understand why this happened, the researchers developed a mathematical framework that treats the tissue as a continuous sheet rather than a collection of individual blocks. They found that this sudden jump is governed by a specific type of mechanical instability, similar to how a bridge might suddenly collapse under a specific load rather than bending slowly. Their analysis showed that the competition between the tension of the cell surfaces and the resistance of the internal cytoplasm creates a tipping point. When the internal resistance is just right, it destabilizes the flat state so thoroughly that the tissue cannot settle into a slightly bent shape; it must jump to a fully folded one to find stability.

This discovery is significant because it shows that the mechanical properties of the cell's interior can qualitatively change the behavior of an entire tissue. It is not just a matter of making the tissue stiffer or softer; it can fundamentally alter the type of folding that occurs. The researchers confirmed that this sudden snapping is not an artifact of their simplified model by testing it with more complex cell shapes, and the effect persisted. They also noted that this sudden jump is rare in nature, with only a few other known examples in physics, such as the buckling of a rod inside a thick elastic medium.

The work suggests that the internal mechanics of cells are just as important as their surfaces in determining how tissues develop and maintain their form. In biological systems, where tissues often fold to create complex shapes like brain folds or the gut, understanding these sudden jumps could be crucial. The researchers emphasize that while their model is a simplification, it reveals a robust phenomenon: the crowded interior of a cell is not a passive bystander. It actively participates in the mechanical decisions of the tissue, capable of turning a gentle bend into a sudden snap. This insight moves the field toward more complete mechanical models of living tissues, where the inside of the cell is treated with the same seriousness as the outside.

Technical Summary: Discontinuous Buckling from Cytoplasmic Mechanics in Vertex Models of Epithelia

Problem Statement
Vertex models of epithelial tissues typically encode cell mechanics through surface terms, specifically the contractile actomyosin cortex and cell adhesion energies. These models often neglect the mechanical contributions of the cytoplasm—the bulk material enclosed by the cortex, which includes the cytosol, organelles (such as the stiff nucleus), and cytoskeletal polymers. While previous studies have noted that surface mechanics alone can lead to non-standard behaviors (such as "unbuckling," where amplitude decreases with compression), the emergent role of cytoplasmic mechanics at the tissue scale remains largely unexplored. This paper investigates how including a minimal representation of cytoplasmic mechanics alters the buckling transition of an epithelial monolayer under external compression.

Methodology
The authors employ a minimal vertex model of an epithelial monolayer consisting of $N$ cells.

Model Definition: Each cell is represented as an isosceles trapezoid with fixed area $a$ . The single-cell energy includes standard surface tension terms (apical, basal, and lateral) and a new cytoplasmic contribution. The cytoplasmic term models the energetic cost of confining cytoplasmic polymers, scaling as $L^{-2}$ with the confinement length $L$ (apical, basal, or lateral). This introduces a cytoplasmic modulus $B$ .
Numerical Simulations: The authors perform numerical minimization of the total energy of the monolayer under varying degrees of relative compression $D$ . They track the buckling amplitude $A$ to determine the nature of the transition (continuous vs. discontinuous).
Analytical Continuum Limit: To explain the numerical findings, the authors derive a continuum theory for $N \gg 1$ . They formulate a Lagrangian for the buckling problem involving the midline tangent angle $\psi(\sigma)$ and the cell stretch $\Lambda$ .
Asymptotic Expansion: Using a double asymptotic expansion for small buckling amplitude ( $A \ll 1$ ) and large $N$ , they derive an effective Landau theory. The expansion is carried out to high orders to capture the competition between surface stretching energy and the emergent bending energy arising from cytoplasmic mechanics.

Key Contributions and Results

Discovery of Discontinuous Buckling: Numerical simulations reveal that the buckling transition, which is typically continuous (second-order) in standard vertex models, becomes discontinuous (first-order) within an intermediate range of the cytoplasmic modulus $B$ . This behavior is reentrant: for both very small and very large values of $B$ , the transition returns to being continuous.
Reentrant Threshold: The critical compression $D^*$ required to trigger buckling exhibits non-monotonic behavior with respect to $B$ , first increasing and then decreasing.
Landau Theory Explanation: The analytical continuum limit successfully reproduces the numerical results. The authors map the problem to a sixth-order Landau theory ( $E = \alpha A^2 + \beta A^4 + \gamma A^6$ $E = α A^{2} + β A^{4} + γ A^{6}$ ).
- The transition between continuous and discontinuous buckling is identified as a tricritical point where the coefficients of the Landau expansion satisfy specific conditions ( $\alpha = \beta = 0$ ).
- The discontinuous regime arises from the competition between the stretching energy of the cell cortices and an emergent bending energy term proportional to the square of the curvature, which originates from the cytoplasmic confinement cost.
Phase Diagram: The authors construct a phase diagram in the space of the scaled cytoplasmic modulus ( $b$ ) and a geometric parameter ( $r$ ). This diagram delineates regions of continuous buckling, discontinuous buckling, and tricritical lines, showing excellent agreement between the asymptotic theory and discrete simulations.
Robustness: The paper demonstrates that the discontinuous buckling is not an artifact of the specific assumption of isosceles trapezoidal cells; a generalized model with arbitrary quadrilateral cells also exhibits this behavior. Furthermore, the phenomenon of "unbuckling" (decreasing amplitude with increased compression) persists even when cytoplasmic mechanics are included.

Significance
The paper establishes that cell-scale mechanics beyond the cortex and adhesion—specifically bulk cytoplasmic effects—can qualitatively alter emergent tissue-scale behaviors. The discovery of a discontinuous buckling transition in a biological tissue model adds to the limited set of known physical systems exhibiting such behavior (e.g., rods in elastic media). By connecting mechanical bifurcations to Landau theory and tricritical points, the work provides a framework for understanding how fluctuations and higher-order mechanical terms influence tissue morphogenesis. The authors conclude that incorporating cytoplasmic mechanics is a necessary step toward developing quantitative, rather than merely qualitative, mechanical models of epithelial tissues and biological shape emergence.

Technical Summary: Discontinuous Buckling from Cytoplasmic Mechanics in Vertex Models of Epithelia

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