---
title: "Membrane mechanics from soft vesicles to active motion"
description: "Learn membrane curvature, Helfrich energy, vesicle shapes, fluctuation measurements, pulling forces and active motion in nine bilingual chapters."
canonical_url: "https://sandboxphysics.com/stories/membrane-mechanics/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/membrane-mechanics/"
last_modified: "2026-10-28"
---

[Membrane mechanics](https://sandboxphysics.com/stories/membrane-mechanics/)

FR20 · An interactive course

# Membrane mechanics from soft vesicles to active motion

Sandbox Physics · 2026-10-28

A membrane only a few molecules thick can enclose a stable shape, tremble under thermal motion and grow a narrow tube under a tiny force. Follow these changes from geometry to measurement, then ask how persistent molecular activity changes the motion.

[Begin with a soft vesicle](https://sandboxphysics.com/stories/membrane-mechanics/curvature-and-constraints/) Open an experiment

A sphere bends in two directions, a cylinder in one and a saddle in opposite directions. Original geometric schematic; no measured data.

## A route from shapes to measurements

Chapters 1–3 introduce curvature, energy and constrained shapes. Chapters 4–5 turn thermal flicker into a stiffness measurement. Chapter 6 follows a pulling experiment and can be read after Chapter 2. Chapter 7 adds composition; Chapters 8–9 use time correlations and published papers to approach active membranes. Each chapter includes derivations, a worked example and exercises with explanations.

Begin with geometry. Add statistical mechanics and mathematical methods when their questions arise; an entire prerequisite route is not required before the first chapter. [Shared textbook foundations](https://github.com/yangzichao/sandboxphysics/issues/27) · [FR19](https://github.com/yangzichao/sandboxphysics/issues/49) · [FR34](https://github.com/yangzichao/sandboxphysics/issues/64)

1. 01 · L1

   ## [How a soft membrane holds a shape](https://sandboxphysics.com/stories/membrane-mechanics/curvature-and-constraints/)

   Use a deflated vesicle to connect curvature, area and volume, calculate reduced volume and distinguish osmotic pressure from tension.

2. 02 · L2

   ## [What it costs to bend a membrane](https://sandboxphysics.com/stories/membrane-mechanics/bending-energy/)

   Derive spherical Helfrich energy and distinguish bending rigidity, tension, area compression modulus and the fixed-topology Gaussian term.

3. 03 · L3

   ## [Several shapes for the same membrane](https://sandboxphysics.com/stories/membrane-mechanics/shape-minima/)

   Compare constrained axisymmetric branches and learn variational minimization, curvature quadrature, convergence and limits of shape inversion.

4. 04 · L2

   ## [Reading bending rigidity from thermal flicker](https://sandboxphysics.com/stories/membrane-mechanics/thermal-modes/)

   Derive quasi-spherical mode variance from equipartition, separate rigidity and tension, and project spherical modes onto an equatorial contour.

5. 05 · L3

   ## [Why a camera can make a membrane look stiffer](https://sandboxphysics.com/stories/membrane-mechanics/camera-and-inference/)

   Derive exposure and angular-pixel transfer, fit rigidity from saved contours, and diagnose noise, focal-depth effects and parameter coupling.

6. 06 · L2

   ## [The force needed to pull a membrane tube](https://sandboxphysics.com/stories/membrane-mechanics/membrane-tethers/)

   Derive long-tube radius and plateau force, include instrument stiffness and infer tension and rigidity from two independent observables.

7. 07 · L3

   ## [How composition and curvature reshape each other](https://sandboxphysics.com/stories/membrane-mechanics/curvature-and-composition/)

   Couple mobile composition to membrane height, derive relaxed mode stiffness and reproduce a finite-wavelength linear instability.

8. 08 · L4

   ## [Distinguishing active motion at the same variance](https://sandboxphysics.com/stories/membrane-mechanics/active-fluctuations/)

   Add a persistent active force, construct equal-variance observations and test an effective-temperature explanation using correlation and response.

9. 09 · L4

   ## [Reading from a tube formula to an active-membrane paper](https://sandboxphysics.com/stories/membrane-mechanics/paper-reading/)

   Freeze paper versions and reproduced scope, recalculate a long-tube reference, then read active-membrane spectra, memory and driver images.

## Three experiments along the way

[01 **Vesicle shapes** Deflate a membrane and compare constrained branches.](https://sandboxphysics.com/labs/vesicle-shapes/) [02 **Membrane flicker** Follow motion through a camera and fit saved contours.](https://sandboxphysics.com/labs/membrane-fluctuations/) [03 **Pull a membrane tube** Drag a probe and separate tube length from instrument motion.](https://sandboxphysics.com/labs/membrane-tether/)

The shape experiment uses a constrained finite axisymmetric family. The fluctuation experiment uses small spherical modes and a defined camera response; its active comparison is a colored-force model. The pulling experiment uses a preexisting long cylinder and a spring-like probe. Parameters are teaching values, and camera records are synthetic.

## 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.

3. [J. Pécréaux et al., Refined contour analysis of giant unilamellar vesicles, European Physical Journal E 13, 277–290 (2004)](https://doi.org/10.1140/epje/i2004-10001-9)

   Contour fluctuation spectroscopy, camera integration and the observation plane. The course constructs its own seeded finite-mode records and derives its own exposure integral. Numerical fits here concern those synthetic records, not the paper’s lipid measurements.

4. [S. A. Rautu et al., The role of optical projection in the analysis of membrane fluctuations, Soft Matter 13, 3480–3483 (2017)](https://doi.org/10.1039/C7SM00108H)

   Equations (1)–(3), Figure 1 and the finite focal-depth measurement problem. The ideal equatorial slice in our model has zero focal depth. Finite angular averaging and exposure are modeled separately; the full optical projection operator is discussed as an extension.

5. [I. Derényi, F. Jülicher and J. Prost, Formation and Interaction of Membrane Tubes, arXiv:cond-mat/0205630v1; Physical Review Letters 88, 238101 (2002)](https://arxiv.org/html/cond-mat/0205630v1)

   Version 1, 29 May 2002, equations (1)–(2): the Helfrich energy and the long cylindrical tube radius and force. We recalculate this analytic limit and add a specified linear instrument spring. The nucleation peak, short-tube profiles, tube interactions and published figure points are outside the reproduced scope.

6. [B. Loubet, U. Seifert and M. A. Lomholt, Effective tension and fluctuations in active membranes, arXiv:1112.2919v2; Physical Review E 85, 031913 (2012)](https://arxiv.org/html/1112.2919v2)

   Version 2, 14 December 2011: quadratic quasi-spherical modes and a direct active force with exponential temporal correlations. Our equal-variance response comparison is an explicitly specified finite linear model. We do not reproduce the paper’s self-consistent excess-area tension curves or permeation and curvature-force cases.

7. [A. Sciortino et al., Active membrane deformations of a minimal synthetic cell, Nature Physics 21, 799–807 (2025)](https://doi.org/10.1038/s41567-025-02839-3)

   Version of Record, 24 March 2025, Figures 2–4 and Data/Code Availability: spatial spectra, correlations and microtubule-driven deformation. The article is CC BY 4.0; raw data and analysis are deposited at doi:10.5281/zenodo.11351857. We provide a reading exercise, not a refit of these experiments or a simulation of their cytoskeleton.

8. [F. Jülicher and R. Lipowsky, Shape transformations of vesicles with intramembrane domains, Physical Review E 53, 2670–2683 (1996)](https://doi.org/10.1103/PhysRevE.53.2670)

   Introduction and domain energy: competition between bending and boundary line energy. Our continuous composition–height Hessian and finite-wave-number numerical example are derived teaching extensions, not a reproduction of the published domain-shape diagrams.