---
title: "Reading from a tube formula to an active-membrane paper"
description: "Freeze paper versions and reproduced scope, recalculate a long-tube reference, then read active-membrane spectra, memory and driver images."
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last_modified: "2026-10-28"
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[Membrane mechanics](https://sandboxphysics.com/stories/membrane-mechanics/)

L4 · Chapter 9

# Reading from a tube formula to an active-membrane paper

Sandbox Physics · 2026-10-28

A reproduction begins with a concrete result: which version, equation or figure, which parameters and which observable. A long-tube analytic limit can be recalculated end to end. An active-membrane experiment needs spatial spectra, temporal correlations and images of its driver read together.

## Before you begin

Long tubes from Chapter 6 and active correlation and response from Chapter 8. Reanalyzing real videos additionally requires contour extraction, microscopy and statistical uncertainty.

- Specify a reproducible result with version, parameters, units, source and limits.
- Distinguish analytic recalculation, synthetic validation and experimental reanalysis when reading an active-membrane paper.

## Start with a result that can be calculated completely

Freeze the 2002 arXiv version 1 by Derényi, Jülicher and Prost. Equation (1) gives energy at specified tension and pressure; equation (2) gives the zero-pressure long-cylinder limit. Our target is equation (2) and independent energy minimization, without digitizing figure points or fitting short-tube shapes.

Use the paper’s illustrative $\kappa=40\,\mathrm{pN\,nm}$ and $\sigma=0.05\,\mathrm{pN\,nm^{-1}}$. Minimize the Chapter 6 energy with respect to radius, then independently evaluate the analytic expressions. They give $r_0=20.0\,\mathrm{nm}$ and $f_0=4\pi\,\mathrm{pN}\simeq12.6\,\mathrm{pN}$.

Reduce tension to one quarter: radius doubles and force halves. Checking absolute values and scaling makes unit or factor errors easier to find than matching a single number.

Source: I. Derényi

## Worked example of a mode-and-exposure benchmark

Take the lowest real shape mode at zero tension and $\kappa=20k_BT$. Chapter 4 gives instantaneous variance $1/480$. At $\lambda T_e=1$, without angular averaging or noise, its exposure-averaged variance is $1/(240e)\simeq0.00153$.

Independently integrate the exponential correlation over the exposure window numerically. Then generate time records with different seeds and inspect whether sample variances scatter around that prediction. The first checks integration; the second checks acquisition and sampling.

An equatorial contour receives other spherical orders at the same contour mode. Finite angular pixels add another transfer factor. State the reduced problem before deciding which values should agree.

Source: B. Loubet · J. Pécréaux et al. · S. A. Rautu et al.

## Separate spatial structure from temporal memory

Sciortino and colleagues’ Version of Record is dated 24 March 2025. Begin with Figure 2’s spatial fluctuations and temporal correlations. Identify whether each vertical axis shows displacement, power or correlation time. Then inspect microtubule organization and local deformation in Figure 3, and traveling deformation in Figure 4.

A large spatial spectrum can reflect lower rigidity, lower tension or stronger driving. Temporal correlation adds persistence, and internal microtubule images provide a further observable. Connecting these measurements is more informative than assigning a mechanism from one spectral slope.

The equal-variance colored-noise example in Chapter 8 helps formulate an identification test. Use it to design correlation–response tests. Reproducing the 2025 experiments additionally requires spatial transport, large deformations and a specified cytoskeletal model.

Source: A. Sciortino et al.

## State where data and figures come from

Examples, schematics and numerical benchmarks in this course are original teaching material. Exported experiment records are seeded synthetic observations. Published-theory recalculation covers specified equations and parameters. The paper’s experimental images, videos and data are not used as our acquired records.

The 2025 article is CC BY 4.0. Its data and analysis scripts are deposited at Zenodo 10.5281/zenodo.11351857. Before reanalysis, inspect the selected files’ licenses, versions, acquisition conditions and processing instructions, and preserve checksums. The article says its membrane simulation model is available from the author on request; open data does not imply every implementation is included.

A complete result identifies input files or seeds, model version, units, selection rules, uncertainty sources and executable commands. State unfinished steps explicitly, such as finite-depth optics, nonlinear domains, full shape equations or cytoskeletal feedback. This lets the next reader continue from the same problem.

Source: A. Sciortino et al.

## Try the reasoning

### [Recalculate the published long-tube reference](https://sandboxphysics.com/labs/membrane-tether/)

1. Use the paper’s parameters, calculate radius and force by hand and compare with the experiment.
2. Reduce tension fourfold and test both scalings. Identify the shapes and force peaks outside this calculation.

### [Preserve a reproducible observation](https://sandboxphysics.com/labs/membrane-fluctuations/)

1. Export one acquisition and retain seed, model version, exposure and pixel count.
2. Separate analytic mode power, power computed from saved contours and fitted parameters. Specify the actual input for an independent rerun.

## Exercises and explanations

1. Does matching the paper’s twenty-nanometer radius reproduce the whole tube paper?

   Read the explanation

   It reproduces the specified long-cylinder analytic limit. Tube nucleation, short profiles, attraction and detached-tube motion require separate models, results and validation.

2. An equilibrium model fits an active spatial spectrum. What should be measured next?

   Read the explanation

   Temporal correlation and independent response to a controlled small force, ideally with simultaneous imaging of the driver. Test whether one parameter set and effective temperature explain them together.

3. What must accompany a paper link before public-data reanalysis?

   Read the explanation

   Dataset version, selected files and checksums, license, units, acquisition conditions and the preprocessing and fitting steps. These details let another reader use the same input.

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

## Sources and calculated scope

1. [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.

2. [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.

3. [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.

4. [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.

5. [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.