Reformulation of Elasticity Theory for Lipid Membranes, with Implications for Gyroid Structures

The elasticity of lipid membranes is generally modeled using the Helfrich free energy, which is expressed in terms of the mean and Gaussian curvatures of the membrane. Here, we suggest a mathematical reformulation of the theory, which writes the free energy as the sum of squares of two modes. These two modes represent the sum and difference of principal curvatures, and are analogous to the splay and Δ deformation modes of nematic liquid crystals [1,2]. This reformulation shows how molecular shape and ordering can induce curved microstructures of lipid membranes. Polar order across the membrane leads to a favored sum of principal curvatures, as has long been recognized by the spontaneous curvature term in Helfrich theory. Likewise, octupolar order within the membrane leads to a favored difference of principal curvatures. In this way, octupolar order provides an explanation for the formation of structures with negative Gaussian curvature, such as gyroids.

[1] J. V. Selinger, Interpretation of Saddle-Splay and the Oseen-Frank Free Energy in Liquid Crystals, Liq. Cryst. Rev. 6, 129 (2018).

[2] J. V. Selinger, Director Deformations, Geometric Frustration, and Modulated Phases in Liquid Crystals, Annu. Rev. Condens. Matter Phys. 13, 49 (2022).