Geometric theory of elastic curves constrained on rigid curved surfaces

Abstract

Slender elastic objects constrained to curved surfaces are ubiquitous in soft systems, with examples ranging from DNA wrapping around histones to actin filaments forming contractile rings on the cell membrane during cytokinesis. In the continuum limit, the conformation of these effectively one-dimensional (1D) objects is governed by both the geometry and mechanics of the embedding surface, and the elasticity of the object. Existing phenomenological typically describe such behavior through elastic energy minimization that incorporates bending and twisting, but neglect stretching that may be important for filaments with finite cross-section or are highly incompatible with the underlying surface. Here we present a coarse-grained geometric theory of 1D elastic curves constrained on rigid surfaces, derived from the first principles. For an isotropic material, our model emerges naturally in terms of the Young's modulus, Poisson's ratio, and area moments of inertia. The resulting effective energy functional explicitly includes stretching in addition to bending and twisting. We minimize it to determine the equilibrium curve conformations on different zero, positive, or negative Gaussian rigid surfaces. Our theory provides a general framework for analyzing the equilibrium configurations of surface-bound elastic curves in biologically and physically relevant settings.