A Gauge-fixing Procedure for Spherical Fluid Membranes and Application to Computations

A distinguishing feature of lipid (bilayer) membranes is their in-plane fluidity caused by free-flowing lipid molecules on the membrane surface. In continuum models for lipid membranes (e.g., the Helfrich–Canham model), fluidity manifests as invariance of the free energy to change in parametrization of the reference surface; a property termed reparametrization invariance. Two different parametric equations of the surface, related through a reparametrization, have identical equilibrium and stability properties. They can therefore be considered equivalent representations of the same surface. Since there are infinitely many ways to parametrize a surface, there are infinitely many equivalent representations for the surface. This highly redundant representation for a surface poses significant challenges to computations. For example, in computational studies using finite element analysis, extreme mesh distortion and spurious zero-energy modes are reported (Feng and Klug, 2006; Ma and Klug, 2008). In this work, by viewing reparametrization invariance as a form of gauge symmetry, we propose a gauge-fixing procedure for the case of topologically spherical membranes. We show that this procedure breaks gauge symmetry and tames the extreme redundancy of the system. We also demonstrate that this procedure is suitable for efficient numerical computations. We obtain accurate equilibrium configurations for the Helfrich–Canham model while circumventing computational issues noted above