Mechanics of axis formation in $\textit{Hydra}$

TL;DR

The paper proposes that Hydra body-axis formation can arise from the coupling of active stresses generated by nematic muscle fibers to the elastic response of a spherical epithelial shell. By modeling the tissue as an active nematic on a curved shell, the authors show that topology constrains force and defect configurations, driving a condensation of active forces at opposite poles and selecting a head-foot axis. They introduce a compact flux-bias parameterization with $oldsymbol{Φ_N = 4π α τ}$ and $oldsymbol{Φ_S = 4π α (1-τ)}$, predicting observable observables such as areal strain, lateral pressure, and normal displacements, as well as the detailed arrangement of $+1$ and $+1/2$ defects at the poles. The framework demonstrates a general physical principle for spontaneous axis specification in morphogenesis, relying on mechanical fields rather than extended morphogen gradients, and provides analytical tools to connect theory with experiments on Hydra and other active tissues.

Abstract

The emergence of a body axis is a fundamental step in the development of multicellular organisms. In simple systems such as $\textit{Hydra}$, growing evidence suggests that mechanical forces generated by collective cellular activity play a central role in this process. Here, we explore a physical mechanism for axis formation based on the coupling between active stresses and tissue elasticity. We analyse the elastic deformation induced by activity-generated stresses and show that, owing to the spherical topology of the tissue, forces globally condense toward configurations in which both elastic strain and nematic defect localise at opposite poles. These mechanically selected states define either a polar or apolar head-food axis. To characterize the condensed regime, we introduce a compact parametrization of of the active force and flux distributions, enabling analytical predictions and direct comparison with experiments. Using this framework, we calculate experimentally relevant observables, including areal strain, lateral pressure, and normal displacements during muscular contraction, as well as the detailed structure of topological defect complexes in head and foot regions. Together, our results identify a mechanical route by which active tissues can spontaneously break symmetry at the organismal scale, suggesting a general physical principle underlying body-axis specification during morphogenesis.

Figures (9)

• Figure 1: (a) Schematic illustration of developing Hydra specimen. The black and gray stripes denote the muscle fibers of the ectoderm. (b,c) Configuration of the muscle fibers of the ectoderm 24h after excision (adapted with permission from Ref. Maroudas:2025). The red and blue frames highlight the future location of the head (North Pole) and foot (South Pole) respectively. Already at this stage of the regeneration process, the two configurations appear topologically distinct, with the ectodermal (endodermal) fibers at the North Pole organized around an aster (a vortex) and those at the South Pole split in two $+1/2$ disclinations. (d,e) Reconstruction of the flux density of the forces actively generated by the muscle fibers in the surrounding of a $+1$ (d) and a pair of $-1/2$ disclinations (e). In the former configuration all contractile active forces condensed at the two poles and the flux density becomes singular. (f,g) The cell areal strain -- i.e. $(A-\langle A \rangle)/\langle A \rangle$ with $A$ the cell area -- versus the distance from either the North (red) or South (blue) poles. (e) Experimental data during stretching events (adapted with permission from Ref. Maroudas:2025) and (f) theoretical predictions.
• Figure 2: Examples of force density fields (yellow arrows) originating from specific defect configurations on the sphere. The gray segments mark the local orientation of the nematic director in proximity of a nematic disclination of strength $s=\pm 1/2,\,\pm 1$.
• Figure 3: Examples of spherical nematic texture (left column) together with their associated active force field and flux density $\varphi$ (right column). (a) Typical configuration of spherical nematics comprising a quartet of isolated $+1/2$ disclinations. The corresponding force field consists of ten defects, of which four of index $\mathop{\mathrm{ind}}\nolimits(v)=0$, colocalized with the nematic disclinations (in the center of the red/blue dumbbell-shaped features) and the remaining six, delocalized. (b) A single $+2$ disclination at the North pole. (c) Condensed state featuring two $+1$ disclinations at the poles. In this case, a uniform active flux across the entire shell is counterbalanced by an equal and opposite flux emanating from the poles.
• Figure 4: (Top) Relaxation dynamics of an active shell obtained from a numerical integration of Eqs. \ref{['eq:displacementshell']} in the limit where $w/R=0$ and $\alpha\lambda<0$. Snapshots are shown at times $t = 0.01,\,0.1,\,0.15$ and $0.75$, from left to right. Topological defects are marked as positive half-charge (red) and negative half-charge (blue). A supplementary animation is available at https://youtu.be/yoFOPmDkAcE. (Bottom) regenerating Hydra under isotonic conditions which exhibits similar coarsening dynamics. (adapted with permission from Ref. Maroudas:2025)
• Figure 5: (a) Temporal evolution of the director's axiality for $\alpha\lambda<0$, defined as the largest eigenvalue $\lambda_1 \in [0, 2/3]$ of the polarity $\bm{P}$ in Eq. \ref{['eq:axiality']}. Snapshots of the nematic defect configurations (insets) are shown at times $t = 0.01,\,0.1,\,0.15$ and $0.75$ from left to right. Perfect axial alignment corresponds to $\lambda_1 = 2/3$ (dashed line), while the classical tetrahedral-symmetry yields $\lambda_1 = 0$. The jump in $\lambda_1$ results from discrete events of defect annihilation. (b) Same quantity for $\alpha\lambda>0$. Insets show the configurations of the director (left) and defects (right) at time $t=0.75$.