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.

Read the flicker

Bending rigidity

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

20

Reduced tension

\bar\sigma

2

Exposure in seconds

T_{\rm exp}

0.04

Contour pixels

N

64

Active variance / thermal variance

A

0

Drag the gold rim marker to inspect another membrane angle. Each slider generates a new saved contour movie.

Acquisition and active-force settings

Active persistence in seconds

3

Random seed

Each movie contains 768 frames, a quarter second apart. The radius is ten micrometres, the bath is 298 kelvin, and viscosity is one millipascal second. Pixel averaging is angular; focus is an ideal equatorial section.

Model and display

The image enlarges radial deviations twofold. The gold marker samples the same saved contour; it applies no force. Playback runs three times faster than physical time. The rigid circle is the mean-radius reference.

The active comparison uses the lowest retained real mode. Its mathematical passive reference increases thermal noise at fixed elasticity and friction to match the theoretical variance. Colored-force exposure is integrated with twelve substeps; the passive exposure is sampled exactly.

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 membraneCamera 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 spectrumFitted camera modelIdeal 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 correlationMatched passive correlationOne 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
Loubet, Seifert and Lomholt · active membrane fluctuations

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 misconceptionA smooth recorded contour proves the membrane is rigid.

Better mental modelExposure 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

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.
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.
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.

Measuring a Flickering Membrane

What the recorded pixels contain

Equal variance can hide a different memory

A saved frame, before fitting

A membrane can act as its own stiffness probe

Core mathematical model

Thermal shape modes

A finite exposure suppresses motion

A force with memory

Common difficulties

A fit needs a measurement model

Run the experiment

Predict what the shutter misses

Separate tension from bending

Compare memory at equal variance

Related experiments and articles

From curvature to active membrane measurements