Small deformations of Helfrich energy minimising surfaces with applications to biomembranes

TL;DR

This paper develops a mathematical framework for small deformations of Helfrich energy-minimizing surfaces, with applications to biomembranes, including theoretical analysis and numerical methods for deformations induced by external forces.

Contribution

It introduces a new model for surface deformations as graphs over original surfaces, deriving Euler-Lagrange equations and providing existence, uniqueness, and computational algorithms.

Findings

Established existence and uniqueness of solutions on spheres and tori.

Developed numerical algorithms for simulating surface deformations.

Numerical examples demonstrating deformation behaviors under various conditions.

Abstract

In this paper we introduce a mathematical model for small deformations induced by external forces of closed surfaces that are minimisers of Helfrich-type energies. Our model is suitable for the study of deformations of cell membranes induced by the cytoskeleton. We describe the deformation of the surface as a graph over the undeformed surface. A new Lagrangian and the associated Euler-Lagrange equations for the height function of the graph are derived. This is the natural generalisation of the well known linearisation in the Monge gauge for initially flat surfaces. We discuss energy perturbations of point constraints and point forces acting on the surface. We establish existence and uniqueness results for weak solutions on spheres and on tori. Algorithms for the computation of numerical solutions in the general setting are provided. We present numerical examples which highlight the behaviour of the surface deformations in different settings at the end of the paper.