---
title: "Vesicle Shapes and Bending Interactive Lab | Sandbox Physics"
description: "How can the same membrane area enclose a long vesicle or a flattened one? A membrane bends into the lowest-energy shape allowed by its area and enclosed volume."
canonical_url: "https://sandboxphysics.com/labs/vesicle-shapes/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/vesicle-shapes/"
last_modified: "2026-10-28"
---

Soft membranes

# Vesicle Shapes and Bending

**How can the same membrane area enclose a long vesicle or a flattened one?** A membrane bends into the lowest-energy shape allowed by its area and enclosed volume. Removing water gives the same membrane more room to fold. A preferred local curvature can then favor a different shape, even when the amount of membrane stays fixed.

Deflate a soft vesicle and compare elongated and flattened shapes with equal area and volume. Color shows local mean curvature; bending energy decides which restricted shape branch is favored.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

An axisymmetric, two-mode radial family minimizes Helfrich bending energy at fixed area and volume. The polar curvature pattern is prescribed. Topology and area stay fixed; neck fission, composition dynamics, protein transport and the complete vesicle phase diagram are outside this model.

Deterministic variational model

## Where the membrane bends

Cyan curve: long branch Amber curve: flat branch

Area, volume and numerical checks

$A/(4\pi R_A^2)=1,\quad v=0.800$

The saved record contains both shapes, model parameters, quadrature resolution and pointwise curvatures. Area and volume are calculated from the same surface used in the scene; this is simulated geometry, with no experimental image.

Physics tutorial

## How a closed membrane chooses its shape

**Background** A lipid bilayer can bend easily while its area changes very little. Letting water leave a closed vesicle gives the same membrane more room to fold.

**Why it matters** An elongated shape and a flattened shape can enclose the same volume with the same area. Comparing their bending energies helps explain why constraints matter as much as softness.

Start with the essentials

- **Focus question**: How can the same membrane area enclose a long vesicle or a flattened one?
- **One-sentence intuition**: Fixed area and volume leave several possible shapes; preferred curvature changes their relative bending cost.

### Core mathematical model

#### Reduced volume measures available excess area

$$
R_A=\sqrt{A/(4\pi)},\qquad v=\frac{V}{4\pi R_A^3/3}\leq1
$$

The reference sphere has the same membrane area. A fully inflated sphere has reduced volume one; losing internal water lowers it.

#### Helfrich bending energy

$$
E=\frac{\kappa}{2}\int(2H-c_0)^2\,dA,\qquad 2H=k_s+k_\phi
$$

The two principal curvatures are positive for an outward-oriented sphere. The Gaussian-curvature term is constant for fixed spherical topology and uniform Gaussian rigidity, so it does not change the branch comparison.

#### The spherical benchmark

$$
c_0=0:\quad E_{\rm sphere}=8\pi\kappa,\qquad\int K\,dA=4\pi
$$

A sphere of any radius has the same pure bending energy. The curvature integral checks the closed topology independently of the selected deformation.

### Common difficulties

#### Recovering stiffness from a still image

**Typical misconception** The shape identifies the membrane bending rigidity.

**Better mental model** In this pure bending problem, uniform rigidity multiplies every candidate energy by the same factor. Shape alone cannot determine it; force or thermal fluctuations provide an independent scale.

#### What a restricted minimum means

**Typical misconception** The lower of two energies proves that the displayed surface is the true equilibrium vesicle.

**Better mental model** The calculation searches a two-mode axisymmetric family. Additional modes, asymmetric buds or invaginations may lower the energy further. The record states the family and numerical resolution.

### Run the experiment

1. 01

   #### Compare two allowed shapes

   Start with A long vesicle, then press Show the other shape. Read the energy difference and compare the curvature curves.

   **What to observe:** Both shapes satisfy the same area and volume. They distribute curvature differently and therefore pay different bending energies.

2. 02

   #### Change the volume at fixed area

   Drag the gold handle down to deflate the membrane, then choose The fully inflated sphere.

   **What to observe:** Deflation opens a choice of elongated and flattened shapes. At maximum volume, the only closed shape with the fixed area is the sphere.

3. 03

   #### Give the polar regions a preference

   Choose Curved polar patches, then reduce Extra polar curvature to zero without moving the other sliders.

   **What to observe:** The favored branch changes from flat to long, with the same area, volume and global preferred curvature. The imposed polar pattern changes the relative energy of the two candidates.

Physical conclusion

Compare shapes under the same geometric constraints before assigning a bending-energy preference.

## Related experiments and articles

Membranes and vesicles · Reading path

### From curvature to active membrane measurements

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How a soft membrane holds a shape**](https://sandboxphysics.com/stories/membrane-mechanics/curvature-and-constraints/)
2. [Article **What it costs to bend a membrane**](https://sandboxphysics.com/stories/membrane-mechanics/bending-energy/)
3. [Article **Several shapes for the same membrane**](https://sandboxphysics.com/stories/membrane-mechanics/shape-minima/)
4. [Article **How composition and curvature reshape each other**](https://sandboxphysics.com/stories/membrane-mechanics/curvature-and-composition/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)