FR20 · An interactive course
Membrane mechanics from soft vesicles to active motion
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.
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 · FR19 · FR34
01 · L1
How a soft membrane holds a shape
Use a deflated vesicle to connect curvature, area and volume, calculate reduced volume and distinguish osmotic pressure from tension.
02 · L2
What it costs to bend a membrane
Derive spherical Helfrich energy and distinguish bending rigidity, tension, area compression modulus and the fixed-topology Gaussian term.
03 · L3
Several shapes for the same membrane
Compare constrained axisymmetric branches and learn variational minimization, curvature quadrature, convergence and limits of shape inversion.
04 · L2
Reading bending rigidity from thermal flicker
Derive quasi-spherical mode variance from equipartition, separate rigidity and tension, and project spherical modes onto an equatorial contour.
05 · L3
Why a camera can make a membrane look stiffer
Derive exposure and angular-pixel transfer, fit rigidity from saved contours, and diagnose noise, focal-depth effects and parameter coupling.
06 · L2
The force needed to pull a membrane tube
Derive long-tube radius and plateau force, include instrument stiffness and infer tension and rigidity from two independent observables.
07 · L3
How composition and curvature reshape each other
Couple mobile composition to membrane height, derive relaxed mode stiffness and reproduce a finite-wavelength linear instability.
08 · L4
Distinguishing active motion at the same variance
Add a persistent active force, construct equal-variance observations and test an effective-temperature explanation using correlation and response.
09 · L4
Reading from a tube formula to an active-membrane paper
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
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
- W. Helfrich, Elastic Properties of Lipid Bilayers: Theory and Possible Experiments, Z. Naturforsch. C 28, 693–703 (1973)
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.
- U. Seifert, K. Berndl and R. Lipowsky, Shape transformations of vesicles, Physical Review A 44, 1182–1202 (1991)
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.
- J. Pécréaux et al., Refined contour analysis of giant unilamellar vesicles, European Physical Journal E 13, 277–290 (2004)
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.
- S. A. Rautu et al., The role of optical projection in the analysis of membrane fluctuations, Soft Matter 13, 3480–3483 (2017)
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.
- I. Derényi, F. Jülicher and J. Prost, Formation and Interaction of Membrane Tubes, arXiv:cond-mat/0205630v1; Physical Review Letters 88, 238101 (2002)
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.
- B. Loubet, U. Seifert and M. A. Lomholt, Effective tension and fluctuations in active membranes, arXiv:1112.2919v2; Physical Review E 85, 031913 (2012)
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.
- A. Sciortino et al., Active membrane deformations of a minimal synthetic cell, Nature Physics 21, 799–807 (2025)
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.
- F. Jülicher and R. Lipowsky, Shape transformations of vesicles with intramembrane domains, Physical Review E 53, 2670–2683 (1996)
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.