Stability and bifurcation analysis in a mechanochemical model of pattern formation

TL;DR

The paper analyzes a one-dimensional mechanochemical pattern formation model where morphogen diffusion couples to tissue elasticity under a global strain constraint, yielding a nonlocal inhibitory term that replaces a classical diffusible inhibitor. It develops a variational framework from an exponential elasticity–morphogen coupling, proves the existence of nonconstant stationary states for small diffusion and uniqueness of the homogeneous state for large diffusion, and conducts a detailed linear and bifurcation analysis. The results show that only unimodal patterns are linearly stable while multimodal patterns are unstable, with both subcritical and supercritical pitchfork bifurcations and fold points generating bistable regimes. This work provides a rigorous mathematical foundation for mechanochemical pattern selection, highlighting how mechanical feedback alone can robustly generate single-peaked patterns with potential relevance to Hydra regeneration and other morphogenetic contexts.

Abstract

We analyze the stability and bifurcation structure of steady states in a mechanochemical model of pattern formation in regenerating tissue spheroids. The model couples morphogen dynamics with tissue mechanics via a positive feedback loop: mechanical stretching enhances morphogen production, while morphogen concentration modulates tissue elasticity. Global strain conservation implements a nonlocal inhibitory effect, realizing a mechanochemical variant of the local activation--long-range inhibition mechanism. For exponential elasticity-morphogen coupling, the system admits a variational formulation. We prove existence of nonconstant steady states for small diffusion and uniqueness of the homogeneous state for large diffusion. Linear stability analysis shows that only unimodal patterns are stable, while multimodal solutions are unstable. Bifurcation analysis reveals subcritical and supercritical pitchforks, with fold bifurcations generating bistable regimes. Our results demonstrate that mechanochemical feedback provides a robust mechanism for single-peaked pattern formation without requiring a second diffusible inhibitor.

Figures (4)

• Figure 2.1: Numerical simulations of model \ref{['equ_varphi']}. Left panel: Emergence of a uni-modal pattern from a small perturbation of the constant stationary solution $U=\kappa$, (example for $D=0.01$, $\kappa=1.5$). Middle panel: Nonconstant stationary solutions for fixed diffusion $D = 10^{-3}$ and varying $\kappa$. Right panel: Nonconstant stationary solutions for fixed $\kappa = 3.0$ and varying $D$. We observe that smaller diffusion leads to more concentrated patterns.
• Figure 2.2: Parameter space analysis and bifurcation structure. (Top) Parameter space $(D, \kappa)$ showing regions of numerical convergence to nonconstant (blue) versus constant (yellow) steady states, with gray areas indicating analytically proven results. (Bottom) Bifurcation diagrams demonstrating subcritical regime with bistability (left, $D < 1/(8\pi^2)$) and supercritical regime with direct pattern formation (right, $D > 1/(8\pi^2)$), where blue and red curves denote stable and unstable branches, respectively.
• Figure 3.1: Schematic representation of the elastic deformation framework for the Hydra spheroid model. Left: Reference configuration $S_0$ parametrized by $g:[0,L] \to S_0$. Right: Deformed configuration $S$ showing the composition of mappings $f =h \circ g$ and the arc-length reparametrization $\tilde{f}$ with $f = \tilde{f} \circ s$.
• Figure 6.1:

Theorems & Definitions (52)

• Proposition 2.1: Global existence and uniqueness
• Remark 2.2: Mass conservation of stationary structures
• Theorem 2.3: Existence of stationary solutions
• Theorem 2.4: Type of stationary solutions
• Corollary 2.5: Existence of multimodal solutions
• Theorem 2.6: Bifurcation structure
• Proposition 4.1
• Remark 4.2
• Remark 4.3
• Lemma 4.4