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Sandbox Physics
Membrane mechanics

L3 · Chapter 5

Why a camera can make a membrane look stiffer

Sandbox Physics ·

During one exposure, a camera averages several membrane positions into one image. Fast fluctuations become smaller. Interpreting that contour as an instantaneous surface often overestimates rigidity. A useful fit describes both membrane dynamics and the camera’s conversion of motion into pixels.

An exposure averages a time interval

Let a real mode q(t)q(t) have equilibrium correlation ⟨q(t)q(0)⟩=C0e−λ∣t∣\langle q(t)q(0)\rangle=C_0e^{-\lambda|t|}. Rate λ\lambda has units of inverse seconds; 1/λ1/\lambda is its memory time. During exposure TeT_e, the camera observes a time average.

The variance of that average sums products from every pair of times in the window. Since correlation depends only on time separation, the double integral reduces to a triangularly weighted integral. A longer window lets positive and negative excursions cancel.

qˉ=1Te∫0Teq(t) dt,⟨qˉ2⟩=2C0Te2∫0Te(Te−t)e−λt dt\bar q=\frac1{T_e}\int_0^{T_e}q(t)\,\mathrm dt,\qquad \langle\bar q^2\rangle=\frac{2C_0}{T_e^2}\int_0^{T_e}(T_e-t)e^{-\lambda t}\,\mathrm dt
The triangular weight counts pairs of times having the same separation inside the exposure.
⟨qˉ2⟩=C0B(λTe),B(x)=2(x−1+e−x)x2\langle\bar q^2\rangle=C_0B(\lambda T_e),\qquad B(x)=\frac{2(x-1+e^{-x})}{x^2}
Short exposures give B→1B\to1; long exposures give B∼2/xB\sim2/x. A small-argument series avoids cancellation when evaluating nearly equal terms numerically.

Source: J. Pécréaux et al.

Contour pixels average in angle

Divide the contour into NN equal angular pixels, each storing mean radius across its width. Averaging a cosine attenuates its amplitude by a sinc factor. Low modes change little; fine waves near the pixel scale attenuate more strongly.

One equatorial mode contains several spherical orders with different relaxation rates. Exposure attenuation belongs inside the spherical-mode sum. Applying a single shutter factor after summing would conflate these rates. Independent localization noise adds its own observed power.

Wm=sinc⁡ ⁣(mπN),sinc⁡(z)=sin⁡zzW_m=\operatorname{sinc}\!\left(\frac{m\pi}{N}\right),\qquad \operatorname{sinc}(z)=\frac{\sin z}{z}
Angular averaging acts on amplitude; power and variance acquire the square of this factor.
Smobs=Wm2∑ℓwℓm2kBTκEℓB(λℓTe)+SmnoiseS_m^{\rm obs}=W_m^2\sum_\ell w_{\ell m}^2\frac{k_BT}{\kappa E_\ell}B(\lambda_\ell T_e)+S_m^{\rm noise}
This observation model combines finite harmonics, a zero-depth equatorial section, angular box pixels and exposure averaging. Finite focal depth requires a different optical operator.

Source: S. A. Rautu et al.

Worked example with one memory time of exposure

Take one independent mode with known relaxation rate, and neglect pixels and localization noise. Set λTe=1\lambda T_e=1. Then B(1)=2/e≃0.736B(1)=2/e\simeq0.736: about seventy-four percent of the instantaneous variance survives.

Suppose true rigidity is 20kBT20k_BT and dimensionless tension is fixed and known. A fit that incorrectly uses instantaneous equipartition interprets the reduced variance as a larger rigidity, 20/0.736≃27.2kBT20/0.736\simeq27.2k_BT. Exposure alone causes roughly a thirty-six percent overestimate in this example.

A multimode fit changes rigidity, tension and relaxation rates together, so its bias need not follow this ratio. The example isolates the mechanism; a complete observation model tests a specified parameter set.

Let saved observations determine the fit

A reproducible synthetic observation saves radial pixels for every frame, timestamps, seed, model version, angular width, exposure, units and truth parameters. The analysis reads pixels, recomputes contour modes and spectra, then fits candidate parameters. Truth is used afterward for comparison and does not enter the estimate.

Applying instantaneous and camera-corrected models to the same saved observations isolates the analysis choice. Repeating acquisition with other seeds reveals finite-sample variation. Exposure, localization noise, harmonic truncation and observation geometry contribute distinct systematic errors.

When tension suppresses all low modes and noise hides the high modes, report coupled parameters instead of every decimal returned by an optimizer. Shorter exposure, better spatial resolution or an independent tension measurement may help more than a longer movie.

Moving surfaceExposure + pixelsSaved contour fit
Surface motion passes through a defined camera response before contours are saved and fitted. Truth parameters provide a separate comparison after fitting. Original measurement schematic.

Try the reasoning

Analyze one contour record two ways

  1. Predict how longer exposure changes visible flicker and rigidity estimated with an instantaneous model.
  2. Save an observation and compare camera-visible contour, observed spectrum, model truth and corrected estimate.
  3. Export the record with seed, exposure and version. Reacquire after changing one measurement condition, and separate instrument bias from random variation.

Exercises and explanations

  1. What is the leading short-exposure correction to B(x)B(x), and why is its sign meaningful?

    Read the explanation

    Expanding the exponential gives B(x)=1−x/3+O(x2)B(x)=1-x/3+O(x^2). The negative sign describes averaging away motion; exposure does not add thermal noise.

  2. Must doubling pixel count halve rigidity error?

    Read the explanation

    No. Pixel averaging is only one contribution. Exposure, localization noise, accessible modes, focal depth and model assumptions also affect the estimate. Compare observed spectra instead of assigning one scaling to resolution.

  3. Someone draws new points from an ideal spectrum and calls them reconstruction of camera observations. What is missing?

    Read the explanation

    A calculation from the saved pixels or contours to those points. Reconstruction must use the actual observations. Independently generated ideal points can serve as a reference.

Sources and calculated scope

  1. 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.

  2. 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.

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