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

TL;DR

This paper develops a continuum mechanics model for active anisotropic plates, capturing how substrate geometry, material anisotropy, and contractility influence cell alignment and deformation, with applications in biology and soft robotics.

Contribution

It introduces a novel nonlinear plate theory incorporating active anisotropic contractions and applies it to predict cell orientation on curved substrates, aligning with experimental observations.

Findings

Predicts bifurcation from perpendicular to parallel cell alignment on cylindrical substrates.

Captures stable orientations on ellipsoidal geometries based on principal curvatures.

Explains divergent cell behaviors on different substrate shapes through a unified model.

Abstract

We develop a general continuum mechanics framework for active anisotropic plates within the F\"oppl-von K\'arm\'an 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.