--- title: "Several shapes for the same membrane" description: "Compare constrained axisymmetric branches and learn variational minimization, curvature quadrature, convergence and limits of shape inversion." canonical_url: "https://sandboxphysics.com/stories/membrane-mechanics/shape-minima/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/membrane-mechanics/shape-minima/" last_modified: "2026-10-28" --- [Membrane mechanics](https://sandboxphysics.com/stories/membrane-mechanics/) L3 · Chapter 3 # Several shapes for the same membrane Sandbox Physics · 2026-10-28 The same area and volume can allow several shapes. A membrane lowers its energy among these possibilities, but an experiment may remain near a local minimum. Interpreting a photograph requires separating geometric constraints, energy minima and the history that brought the vesicle there. ## Before you begin Geometry and Helfrich energy from Chapters 1–2, plus the idea of a function minimum. We introduce numerical optimization over a few shape parameters. Full shape equations require variational calculus and differential equations. - Explain how area and volume remove variational freedom, and design residual and convergence checks. - Explain why equilibrium shape alone cannot determine bending rigidity. ## Choose a tractable family of surfaces Choose a symmetry axis and measure polar angle $\theta$ from its north pole. The radial distance $r(\theta)$ defines the surface. Two low-order Legendre polynomials provide shape freedom: the second-order term mainly distinguishes elongated and flattened shapes, and the fourth-order term adjusts their contour. The exponential keeps the radius positive. Scale $s$ controls area; $a$ and $b$ control shape. This family is smooth, axisymmetric and star-shaped about its center, so each radial ray crosses the membrane once. Indentations with multiple radial crossings, narrow-neck limits and asymmetric shapes lie outside this family. Its restricted elongated and flattened branches provide a controlled energy comparison. $$ r(\theta)=s\exp\!\left[aP_2(\cos\theta)+bP_4(\cos\theta)\right] $$ Here $P_2(x)=(3x^2-1)/2$ and $P_4(x)=(35x^4-30x^2+3)/8$. Shape coefficients are dimensionless and $s$ has length units. Source: U. Seifert · W. Helfrich ## Enforce area and volume before comparing energy Temporarily set the overall scale to one and integrate area and volume over polar angle. Rescale the surface to the target area. Uniform scaling leaves reduced volume unchanged, so finding the requested reduced volume is now a problem in shape coefficients. Fix $b$ and search separately for positive and negative $a$. Bisection evaluates the interval midpoint, checks which side of the target reduced volume it lies on and retains the half that brackets the solution. Only after finding admissible shapes can we compare energies. Minimize each branch over a finite range of fourth-order coefficients. The result is a constrained minimum within this family and search range. Finding the physical global minimum requires a larger shape space and checks for self-contact and instability. $$ A=2\pi\int_0^\pi r\sin\theta\sqrt{r^2+r_\theta^2}\,\mathrm d\theta,\qquad V=\frac{2\pi}{3}\int_0^\pi r^3\sin\theta\,\mathrm d\theta $$ Area sums the circumference times arclength of small rings; volume integrates radial sectors over solid angle. $$ s=\sqrt{\frac{A_{\rm target}}{A_{s=1}}},\qquad v(a,b)=v_{\rm target} $$ The first equation eliminates scale and the second removes one shape degree of freedom. Optimization follows the remaining constrained curve. ## Worked example of a constraint residual For $R_A=10\,\mu\mathrm m$ and $v=0.80$, target area is $1.26\times10^3\,\mu\mathrm m^2$ and target volume is approximately $3.35\times10^3\,\mu\mathrm m^3$. A relative volume residual of $10^{-4}$ means an error of about $0.335\,\mu\mathrm m^3$. First ask whether the candidate actually solves the requested problem. Then double angular quadrature resolution and compare energy, reduced volume and the Gaussian-curvature integral. Convergence supports attributing a persistent difference to the model instead of a coarse discretization. The sphere supplies independent references: at zero spontaneous curvature, mean-curvature energy divided by $\kappa$ approaches $8\pi$ and integrated Gaussian curvature approaches $4\pi$. These test local curvature and global closed-surface geometry. ## Which scale is missing from a photograph? Suppose only mean-curvature energy matters and area, volume and spontaneous curvature are specified. Multiplying $\kappa$ by any positive constant multiplies every candidate energy without moving the minimum. Even a perfect contour therefore cannot determine bending rigidity on its own. Unknown tension, pressure and protein distributions can introduce further compensations. A fitted number is an answer within the chosen assumptions. Thermal fluctuations, a known applied force or an independent pressure measurement can add information absent from shape alone. This explains why quite different material parameters may give almost indistinguishable contours. A useful inversion first asks which parameters affect the available observables, then chooses a complementary experiment. Thermal motion will provide an independent energy scale in the next part of the course. $$ F_b=\kappa\,\mathcal E[\text{shape}],\qquad \arg\min F_b=\arg\min\mathcal E\quad(\kappa>0) $$ This argument requires rigidity to appear as an overall positive factor. Additional independent energy scales require a new identifiability analysis. ## Try the reasoning ### [Compare two branches at the same constraints](https://sandboxphysics.com/labs/vesicle-shapes/) 1. Choose moderate deflation and predict whether elongated and flattened surfaces can share area and volume. 2. Compare their area, volume and energy, then vary spontaneous curvature and look for a change in ranking. 3. Enable the prescribed polar curvature preference. Explain the contour change, then identify the extra equation needed if curvature-generating proteins could move. ## Exercises and explanations 1. Two energies are close, but the relative volume residual exceeds the relative energy difference. What should improve first? Read the explanation Tighten the geometric constraints and improve quadrature. Different volumes compare different physical problems. Discuss a small energy difference only after controlling constraint and discretization errors. 2. Can a minimum in a finite parameter family be called a stable vesicle? Read the explanation It is a constrained minimum in that family. More general stability requires allowing omitted perturbations, checking second-order energy changes, self-contact and the model’s range of validity. 3. Design a second measurement that determines rigidity when shape alone cannot. Read the explanation Record thermal fluctuations at known temperature and fit rigidity and tension after camera correction, or measure both tube radius and pulling force. Each supplies an independent energy or force scale. [Previous: What it costs to bend a membrane](https://sandboxphysics.com/stories/membrane-mechanics/bending-energy/) [Next: Reading bending rigidity from thermal flicker](https://sandboxphysics.com/stories/membrane-mechanics/thermal-modes/) ## Sources and calculated scope 1. [W. Helfrich, Elastic Properties of Lipid Bilayers: Theory and Possible Experiments, Z. Naturforsch. C 28, 693–703 (1973)](https://doi.org/10.1515/znc-1973-11-1209) Original curvature-elasticity paper, pages 693–703. We use its local quadratic curvature energy with an explicitly stated sphere-positive curvature convention. The sphere, tube and small-slope examples are derived here; no original figure or measured material parameter is copied. 2. [U. Seifert, K. Berndl and R. Lipowsky, Shape transformations of vesicles, Physical Review A 44, 1182–1202 (1991)](https://doi.org/10.1103/PhysRevA.44.1182) Published paper: spontaneous-curvature and bilayer-coupling ensembles and axisymmetric shape branches. Our shape experiment minimizes within a finite radial variational family at fixed area and volume. It does not reproduce the complete published phase diagram, neck limits or stomatocytes.