---
title: "Measuring a Flickering Membrane Interactive Lab | Sandbox Physics"
description: "Can a camera tell how hard a membrane is to bend? Thermal motion bends a soft membrane more strongly than a stiff one."
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last_modified: "2026-10-28"
---

FR20 · Fluctuation spectroscopy

# Measuring a Flickering Membrane

**Can a camera tell how hard a membrane is to bend?** Thermal motion bends a soft membrane more strongly than a stiff one. Fitting how the fluctuations change with wavelength can reveal bending rigidity, provided the fit accounts for tension and for the small, fast ripples lost during imaging.

Watch a membrane flicker, then recover its bending rigidity from saved contour frames. Slow the shutter and compare thermal motion with persistent active forcing.

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

Model assumptions

A finite quasi-spherical model retains spherical harmonics through degree twelve. Tension controls the mean excess area; area and volume are not enforced on each linearized frame. The optical section is exactly equatorial; finite focal depth and localization noise are omitted. Angular pixel averaging and exposure affect the saved contour. Active forcing is a linear colored-noise model. All records are synthetic; there is no fit to a particular lipid or cell.

Simulated membrane Camera samples

Equatorial section · radial deviations enlarged twofold

Model rigidity

$\kappa/(k_{\rm B}T)$

$20$

Camera-corrected estimate

$\widehat\kappa/(k_{\rm B}T)$

$20$

Estimate ignoring camera

$\widehat\kappa_{\rm naive}/(k_{\rm B}T)$

$20$

Estimated reduced tension

$\widehat{\bar\sigma}$

$20$

## What the recorded pixels contain

Saved contour spectrum Fitted camera model Ideal thermal reference

$m,\qquad \tfrac12\langle a_m^2+b_m^2\rangle$

Both axes are logarithmic. Dots come from a Fourier transform of the saved radial pixels. The fit varies rigidity and tension together; optics are known.

## Equal variance can hide a different memory

Active correlation Matched passive correlation One minus response

$t\;[\mathrm s],\qquad C(t)/C(0),\quad 1-K\chi(t)$

Lines are the defined linear-model references. Dots are correlations of stored instantaneous mode samples. The passive reference is heated to have the same theoretical variance as the active mode.

## A saved frame, before fitting

Import JSON

$\varphi\;[\mathrm{rad}],\qquad [r(\varphi)-R]/R$

Cyan is the simulated instantaneous contour at the end of exposure. Amber is the exposure-averaged pixel record. Only amber enters the stiffness fit. Export keeps every radial pixel, sample time, seed, parameters, units and model version.

Fit range, response and numerical limits

$X(\omega)=\frac{\omega S_x(\omega)}{2k_{\rm B}T_{\rm eff}\operatorname{Im}\chi(\omega)}$

The ratio at the lowest mode’s relaxation frequency is $1$. The effective temperature matches the variance. Equilibrium gives one at all frequencies; colored forcing can make the ratio frequency dependent.

[Pécréaux and colleagues · contour analysis and exposure](https://doi.org/10.1140/epje/i2004-10001-9)
[Loubet, Seifert and Lomholt · active membrane fluctuations](https://arxiv.org/abs/1112.2919)

Physics tutorial

## A membrane can act as its own stiffness probe

**Background** Watch a membrane flicker, then recover its bending rigidity from saved contour frames. Slow the shutter and compare thermal motion with persistent active forcing.

**Why it matters** Thermal energy keeps pushing the membrane out of shape. A stiff membrane resists short ripples, so a movie contains information about its elasticity.

Start with the essentials

- **Focus question**: Can a camera tell how hard a membrane is to bend?
- **One-sentence intuition**: To infer stiffness from a movie, predict how the camera changes each fluctuating mode.

### Core mathematical model

#### Thermal shape modes

$$
\langle a_{\ell m}^2\rangle=\frac{1}{\kappa_* (\ell-1)(\ell+2)[\ell(\ell+1)+\bar\sigma]},\qquad \kappa_*=\frac{\kappa}{k_{\rm B}T}
$$

The coefficients multiply orthonormal real spherical harmonics. Translation and mean radius are removed. Bending dominates at high degree; tension affects longer waves.

#### A finite exposure suppresses motion

$$
B(x)=\frac{2(x-1+e^{-x})}{x^2},\qquad x=\lambda_\ell T_{\rm exp}
$$

Each passive mode has an exponential time correlation. Averaging its displacement during an exposure multiplies its variance by this factor. Fast modes lose the most power.

#### A force with memory

$$
\dot x=-\lambda x+f+\xi_{\rm th},\qquad \dot f=-f/\tau_a+\xi_a
$$

The added force has its own persistence time. The comparison passive mode is heated to match the active mode’s variance; its mechanical response stays the same.

### Common difficulties

#### A fit needs a measurement model

**Typical misconception** A smooth recorded contour proves the membrane is rigid.

**Better mental model** Exposure and pixel averaging also smooth the contour. The corrected fit uses the known camera settings and the radial pixels saved in the record.

### Run the experiment

1. 01

   #### Predict what the shutter misses

   Compare A flickering membrane and The camera smooths motion. Inspect the high-mode end of the saved spectrum.

   **What to observe:** Long exposures reduce the amber contour’s rapid ripples. Ignoring the camera can make the same membrane look stiffer.

2. 02

   #### Separate tension from bending

   Choose Tension quiets long waves, then move Reduced tension while keeping Bending rigidity fixed.

   **What to observe:** The broad distortions shrink first. Fitting several mode numbers helps separate tension from bending rigidity.

3. 03

   #### Compare memory at equal variance

   Choose Equal width, different memory. Compare the saved mode correlations and the response reference below.

   **What to observe:** The active mode keeps a longer memory although the comparison has the same theoretical variance. A single effective temperature cannot match its correlation and response at every frequency.

Physical conclusion

A spectrum reveals elasticity only after the optics and the source of fluctuations are specified.

## 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 **Reading bending rigidity from thermal flicker**](https://sandboxphysics.com/stories/membrane-mechanics/thermal-modes/)
2. [Article **Why a camera can make a membrane look stiffer**](https://sandboxphysics.com/stories/membrane-mechanics/camera-and-inference/)
3. [Article **Distinguishing active motion at the same variance**](https://sandboxphysics.com/stories/membrane-mechanics/active-fluctuations/)
4. [Article **Reading from a tube formula to an active-membrane paper**](https://sandboxphysics.com/stories/membrane-mechanics/paper-reading/)
5. [Lab **Pulling a Membrane Tether** Recalculate the published long-tube reference](https://sandboxphysics.com/labs/membrane-tether/)

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

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