Simplified analytical framework for fundamental vesicle shapes

Membrane structures play a central role in cellular processes, adopting morphologies such as sheetlike cisternae of the endoplasmic reticulum and mitochondrial cristae, tubular extensions of the ER and neurons, and cup-shaped double membranes of autophagosomes. Theoretical descriptions of these fundamental geometries typically require solving singular nonlinear shape equations or performing computationally intensive coarse-grained and triangulated vesicle simulations. Here, we introduce a simple analytical framework that provides closed-form expressions for the key shape properties of tubes, sheets, and cups, including volume-to-area ratio, bending energy, and asymmetry. This approach eliminates the need for complex mathematical or numerical methods while retaining quantitative accuracy. We validate and modify our results by direct comparison with accurate results form Monte Carlo minimizations of triangulated vesicles. The framework thus offers a fast and accessible method for estimating membrane shape energetics, providing a practical tool for studies of cellular and biomimetic membrane systems.