It is with great regret that Prof. Wolfgang Helfrich passed away on 28 September 2025 in Berlin. As the founder of the membrane liquid crystal model, Prof. Helfrich made outstanding contributions to membrane physics and liquid crystal display technology. This review article is written in his memory. Biomembranes, primarily composed of lipid bilayers, are not merely passive barriers but dynamic and complex materials whose shapes are governed by the principles of soft matter physics. This review explores the shape problem in biomembranes through the lens of material science and liquid crystal theory. Beginning with classical analogies to crystals and soap bubbles, it details the application of the Helfrich elastic model to explain the biconcave shape of red blood cells. The discussion extends to multi-layer systems, drawing parallels between the focal conic structures of smectic liquid crystals, the geometries of fullerenes and carbon nanotubes, and the reversible transitions in peptide assemblies. Furthermore, it examines icosahedral self-assemblies and shape formation in two-dimensional lipid monolayers at air/water interfaces. At the end of the paper, we find that the shapes such as cylinders, spheres, tori, bicocave discoids and Delaunay surfaces form a group. This result is merely an intrinsic geometric feature of these shapes and is independent of the biomembrane equation. When the pressure on the membrane, surface tension, and bending modules meet certain conditions, the biomembrane will take on these shapes. The review concludes by highlighting the unifying power of continuum elastic theories in describing a vast array of membrane morphologies across biological and synthetic systems.
Deep Dive into The Beauty of Mathematics in Helfrich's Biomembrane Theory.
It is with great regret that Prof. Wolfgang Helfrich passed away on 28 September 2025 in Berlin. As the founder of the membrane liquid crystal model, Prof. Helfrich made outstanding contributions to membrane physics and liquid crystal display technology. This review article is written in his memory. Biomembranes, primarily composed of lipid bilayers, are not merely passive barriers but dynamic and complex materials whose shapes are governed by the principles of soft matter physics. This review explores the shape problem in biomembranes through the lens of material science and liquid crystal theory. Beginning with classical analogies to crystals and soap bubbles, it details the application of the Helfrich elastic model to explain the biconcave shape of red blood cells. The discussion extends to multi-layer systems, drawing parallels between the focal conic structures of smectic liquid crystals, the geometries of fullerenes and carbon nanotubes, and the reversible transitions in peptide
In 1922, French mineralogist Georges Friedel, with his son, systematically laid the experimental foundations [1] of liquid crystal science. His core contributions were threefold: First, he established the fundamental phase classification, defining the layered "smectic" and the orientationally ordered "nematic" phases. Second, using polarizing microscopy, he provided the first detailed description of the complex focal conic texture in smectics, recognizing it as an intrinsic feature of the layered structure. Third, although not using the term "Dupin cyclides," his precise description of the focal conics (an ellipse and a confocal hyperbola) provided the essential experimental groundwork for all future geometric and theoretical explanations.
Building directly upon Friedel’s observations, William Bragg, in his seminal paper “The Focal Conic Structure in Smectic Liquid Crystals” [2], provided the critical geometric insight. He demonstrated that the seemingly complex focal conic texture could be described elegantly by a family of surfaces maintaining constant interlayer spacing. Bragg proved that the mathematical solution to this condition is a family of surfaces known as Dupin cyclides, explicitly identifying them as the geometric essence of the smectic layer arrangement. This work decisively answered the “how” by connecting the microscopic defects to a precise, universal geometric model. While Bragg explained the geometry, the fundamental “why”-the underlying energy minimization principle-was later explained using continuum elasticity theory. The crucial framework for this was established by Wolfgang Helfrich in 1973. In his paper “Elastic properties of lipid bilayers: theory and possible experiments” [3], he formulated a general curvature elastic energy for fluid membranes. This Helfrich free energy, expressed in terms of mean and Gaussian curvature, created a unified continuum theory for all thin film systems governed by curvature elasticity. This model was directly applied to smectic liquid crystals, proving that the Dupin cyclide configuration described by Bragg is indeed the energy-minimizing solution under the constraint of layer incompressibility, thereby completing the theoretical picture from phenomenon to geometry to energy.
The collaboration between Ou-Yang Zhong-Can and Wolfgang Helfrich in the late 1980s produced two foundational papers that bridged the gap between the theory of membrane elasticity and the prediction of complex biological shapes. Their key publications are “In-stability and deformation of a spherical vesicle by pressure” [4], “Bending energy of vesicle membranes: General expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders” [5].
Building upon Helfrich’s curvature elasticity model for membranes, their rigorous application of variational methods led to the derivation of a universal differential equation governing vesicle equilibrium shapes. This pivotal result, known as the “Zhong-Can-Helfrich equation,” provided the first quantitative theoretical framework for explaining intricate biomembrane morphologies, most notably the biconcave disc shape of red blood cells.
For over a century, the unique biconcave disc shape of the human red blood cell has been a major puzzle in biophysics. The analysis is built upon the Helfrich theory of fluid membrane. The lipid bilayer is treated as a two-dimensional liquid crystal sheet, with its equilibrium shape determined by minimizing the bending elastic energy. Naito, Okuda, and Ou-Yang’s primary contribution [6] was to demonstrate that under the condition of zero spontaneous curvature, the axisymmetric shape equation admits a specific analytic solution that perfectly describes the classic biconcave disc profile. The work was also profoundly predictive, forecasting novel shapes like toroidal vesicles that were subsequently verified by experiment.
Beyond biophysics, Ou-Yang later expanded his research to elucidate the complex forms found in diverse soft matter systems, including focal conic domains in liquid crystals, the structure of carbon nanotubes, and the assembly of viral capsids.
Prof. Ou-Yang et al. [7], [9] have developled Helfrich theory in fullerenes and carbon nanotubes. They take the continuum limit of Lenosky’s discrete carbon network model to formulate a curvature elasticity theory in the same spirit as the Helfrich model in its mathematical form. Both embody the physical idea of using continuous curvature to describe microscopic interactions.
The morphology of nanofiber membranes or vesicles formed by the self-assembly of peptide amphiphiles is similarly governed by a balance between curvature elastic energy (driven by the molecules’ intrinsic spontaneous curvature) and surface energy. The Helfrich model [8] serves as the natural theoretical tool for analyzing their stability.
Over the past century, numerous scientists have made outstanding contributions in these
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