The mechanics of anisotropic active plates with applications to cell alignment on curved substrates

TL;DR

The paper develops a general 3D hyperelastic active-body framework with anisotropic reinforcement, then reduces it to a nonlinear Föppl–von Kármán plate model using a multiplicative decomposition $\mathbb{F}=\mathbb{F}_{\rm e}\mathbb{F}_{\rm a}$ and a transversely isotropic energy. It derives coupled PDEs for the out-of-plane deflection $\xi$ and the in-plane Airy potential $\phi$, highlighting how active distortions and anisotropy generate bending moments and curvature-targeting terms $C_m$, $\mathbf{b}_{\rm a}$, and $\mathbb{W}_{\rm a}$. The authors apply the framework to curvature-driven cell alignment on cylindrical and ellipsoidal substrates, predicting orientation bifurcations and stable oblique states that qualitatively reproduce experimental observations across cell types, including dependence on active contractility $\alpha$, substrate curvature $\kappa_1$, and material stiffness ratios $\mu/2k_{44}$. The results illuminate how bending resistance, anisotropic stiffness, and thickness-dependent activity jointly determine cellular orientation on curved substrates and generalize to thin active fiber-reinforced structures in soft robotics, morphogenesis, and tissue engineering.

Abstract

We develop a general continuum mechanics framework for active anisotropic plates within the Föppl-von Kármán limit, incorporating a preferential direction and inelastic active contractions in geometrically nonlinear plate theory. Through asymptotic expansion, we derive coupled equilibrium equations for plates with transversely isotropic and possibly inhomogeneous reinforcement undergoing spatially varying active contractions through their thickness. The framework highlights the coupling between material anisotropy and active deformations, with target curvatures that compete with imposed geometric constraints. To demonstrate its capabilities, we apply the model to curvature-induced cell alignment, where substrate geometry, cytoskeletal anisotropy, and contractility interact to determine orientation. For cylindrical substrates, the model predicts a supercritical bifurcation in preferred orientation, from perpendicular to parallel, through an oblique orientation governed by the ratio of active contractility to substrate curvature and modulated by material stiffness. For ellipsoidal geometries, we capture stable parallel, perpendicular, and oblique configurations depending on principal curvatures, whereas spherical substrates do not show preferred alignments. These predictions qualitatively reproduce experimental observations across cell types, explaining divergent behaviors between contractile epithelial cells and stiffer fibroblasts in a rigorous context. Beyond cellular mechanics, this framework applies broadly to thin fiber-reinforced active structures in soft robotics, morphogenesis, and tissue engineering.

Figures (5)

• Figure 1: Schematic of the reference configuration $\mathcal{B}_r$ of the plate and of the decomposition $\mathbb{F} = \mathbb{F}_{\rm e}\mathbb{F}_{\rm a}$. The middle surface of the plate is denoted by $\mathcal{S}$.
• Figure 2: Sketch of the mechanical setup for a cell adhering to a cylindrical surface.
• Figure 3: (a): Bifurcation diagram showing stable (solid lines) and unstable (dashed lines) orientations on a cylinder, as a function of $\alpha/\kappa_1$, for a fixed value of $\mu/2k_{44}= 0.3$. The bifurcation point $\Lambda_c$ is defined in \ref{['eq:bif_point_cylinder']}. (b): Bifurcation diagram in the space $(\alpha/\kappa_1, \cos^2\theta)$ for the cylinder case and different values of $k_{44}$.
• Figure 4: (a) Plot of $\cos^2(\theta)$ as a function of $\Lambda = \alpha/\kappa_1$ for the ellipsoid case, with $A = 2$ and $B = 1$. (b): Plot of the stationary angles $\theta$ as a function of $\Lambda = \alpha/\kappa_1$ for different values of $A/B$. The dashed and solid lines describe unstable and stable configurations, respectively.
• Figure 5: Plots of the normalized bending energy $\mathscr{W}_{\rm bend}$ as a function of the angle, for different values of the ratio $A/B$. (a): $\alpha = 10^{-3} \mu$m$^{-1}$. (b): $\alpha = 3\cdot10^{-3} \mu$m$^{-1}$. (c): $\alpha = 5\cdot10^{-3} \mu$m$^{-1}$. (d): Comparison for different values of $\alpha$ and $A/B$. For each value of $\alpha$, darker curves correspond to $A/B = 2$, while lighter curves correspond to $A/B = 20$.