arxiv: 2511.00775 · v3 · submitted 2025-11-02 · ❄️ cond-mat.soft · q-bio.QM

Detecting active L\'evy particles using differential dynamic microscopy

Mingyang Li , Yu'an Li , H. P. Zhang , Yongfeng Zhao This is my paper

Pith reviewed 2026-05-18 01:57 UTC · model grok-4.3

classification ❄️ cond-mat.soft q-bio.QM

keywords Lévy walksdifferential dynamic microscopyactive particlescell motilityE. coliE. gracilispower-law distributionsmicroorganism motion

0 comments p. Extension

The pith

Differential dynamic microscopy distinguishes active Lévy motion in E. gracilis from E. coli.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends differential dynamic microscopy to self-propelled particles with run-time distributions that follow power-law tails. Validation on synthetic image sequences shows the method extracts the expected scaling signatures when data covers length scales roughly ten times larger than the persistence length and speed variation stays moderate. Application to real videos then indicates that E. coli trajectories lack the Lévy signature while E. gracilis motion matches the active Lévy particle model more closely.

Core claim

We extend the differential dynamic microscopy method to self-propelled Lévy particles whose run-time distribution has an algebraic tail. Validation on synthetic imaging data shows that reliable detection requires accessing length scales an order of magnitude larger than its persistence length when speed variability is moderate. Applying the protocol to experimental data of E. coli and E. gracilis, we find that E. coli does not exhibit a signature of Lévy walks, while E. gracilis is better described as active Lévy particles.

What carries the argument

Extended differential dynamic microscopy protocol that extracts power-law scaling from image correlation functions for active particles with algebraically tailed run times.

If this is right

Synthetic data confirms the protocol recovers Lévy signatures when the scale requirement is met.
E. coli motion statistics are inconsistent with Lévy walks under the tested conditions.
E. gracilis motion statistics align with the active Lévy particle description.
The distinction holds only for datasets that reach length scales an order of magnitude beyond the persistence length.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same image-analysis route could classify Lévy-like search in additional motile organisms without needing full trajectory reconstruction.
If the E. gracilis classification holds at still larger scales, it points to a different long-range exploration strategy than E. coli employs.
Adaptations of the protocol might be tested on other active-matter systems where power-law run times are suspected.

Load-bearing premise

The experimental datasets must cover length scales an order of magnitude larger than the persistence length when speed variability remains moderate.

What would settle it

Direct tracking of run times in E. gracilis at scales much larger than the persistence length that fails to show an algebraic tail, or similar data for E. coli that does show one, would falsify the reported distinction.

Figures

Figures reproduced from arXiv: 2511.00775 by H. P. Zhang, Mingyang Li, Yongfeng Zhao, Yu'an Li.

**Figure 2.** Figure 2: FIG. 2. The intermediate scattering functions (ISFs) of active L´evy particles (ALPs) and their asymptotic behavior. We [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. The mean-squared displacement (MSD) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. Validation of the fitting protocol using simulations. The ISFs calculated from the synthetic images are shown in [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Fitting results of the ISFs of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. The ISFs of [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

read the original abstract

Detecting L\'evy flights of cells has been a challenging problem in experiments. The challenge lies in accessing data in spatiotemporal scales across orders of magnitude, which is necessary for reliably extracting a power-law scaling. Differential dynamic microscopy has been shown to be a powerful method that allows one to acquire statistics of cell motion across scales, which is a potentially versatile method for detecting L\'evy walks in biological systems. In this article, we extend the differential dynamic microscopy method to self-propelled L\'evy particles, whose run-time distribution has an algebraic tail. We validate our protocol using synthetic imaging data and show that a reliable detection of active L\'evy particles requires accessing length scales of an order of magnitude larger than its persistence length, if the variability in particle speed is moderate. Applying the protocol to experimental data of E. coli and E. gracilis, we find that E. coli does not exhibit a signature of L\'evy walks, while E. gracilis is better described as active L\'evy particles.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends DDM to active Lévy particles with a clear scale requirement from synthetic tests, then claims E. coli lacks the signature while E. gracilis fits it, but that distinction depends on whether the real datasets actually reach the needed length scales.

read the letter

The main point is that they have worked out a DDM protocol for spotting algebraic run-time tails in self-propelled particles and backed it with synthetic imaging data. The tests show reliable power-law extraction needs length scales roughly ten times the persistence length when speed variation stays moderate. They then run the same protocol on E. coli and E. gracilis movies and report that only the latter matches active Lévy behavior.

Referee Report

2 major / 2 minor

Summary. The manuscript extends differential dynamic microscopy (DDM) to self-propelled Lévy particles whose run-time distribution has an algebraic tail. Synthetic imaging data are used to validate the protocol and establish that reliable detection of the power-law signature requires accessing length scales an order of magnitude larger than the persistence length when speed variability is moderate. Application to experimental DDM data then concludes that E. coli lacks a Lévy-walk signature while E. gracilis is better described as active Lévy particles.

Significance. If the central claim holds, the work supplies a practical DDM-based protocol for identifying Lévy walks in cell motility, exploiting DDM’s ability to gather statistics across wide spatiotemporal scales. The synthetic validation is a clear strength, as it supplies a concrete, testable scale requirement rather than an ad-hoc criterion. The biological distinction between the two organisms, however, rests on whether the real datasets satisfy that requirement relative to their fitted persistence lengths.

major comments (2)

[experimental results] § on experimental application (following the synthetic validation): the distinction that E. coli shows no Lévy signature while E. gracilis fits the active-Lévy model is load-bearing for the paper’s main conclusion, yet the manuscript does not report the fitted persistence lengths for each organism nor confirm that the experimental 1/q range exceeds those lengths by the factor of ~10 required by the synthetic tests under moderate speed variability. If the accessed scales fall short, the reported absence for E. coli could be an artifact of insufficient dynamic range.
[validation] Synthetic-validation section: the protocol’s robustness to post-processing choices (e.g., image segmentation thresholds, background subtraction, or fitting windows) is not quantified; because the final organism classification depends on these steps, an explicit sensitivity analysis or error-propagation estimate is needed to support the claim that the method reliably extracts the power-law signature.

minor comments (2)

[methods] Notation for the run-time distribution (e.g., the exponent of the algebraic tail) should be introduced once in the methods and used consistently in figures and text.
[figures] Figure captions for the synthetic-data panels should state the exact speed-variability parameter and the ratio of probed length scale to persistence length so readers can directly compare with the experimental q-range.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us strengthen the presentation of our results. We address each major comment below and have revised the manuscript to incorporate the requested information and analyses.

read point-by-point responses

Referee: [experimental results] § on experimental application (following the synthetic validation): the distinction that E. coli shows no Lévy signature while E. gracilis fits the active-Lévy model is load-bearing for the paper’s main conclusion, yet the manuscript does not report the fitted persistence lengths for each organism nor confirm that the experimental 1/q range exceeds those lengths by the factor of ~10 required by the synthetic tests under moderate speed variability. If the accessed scales fall short, the reported absence for E. coli could be an artifact of insufficient dynamic range.

Authors: We agree that an explicit comparison between the experimental length scales and the persistence lengths obtained from the fits is necessary to validate the conclusions against the synthetic benchmark. Although the accessed q-range (and thus length scales) was stated in the experimental methods, the fitted persistence lengths were not directly tabulated or compared to the factor-of-10 criterion. In the revised manuscript we have added this comparison in the experimental-results section, including the fitted persistence lengths for both organisms and a statement confirming that the experimental dynamic range satisfies the requirement identified in the synthetic tests for moderate speed variability. With this information included, the absence of a Lévy signature in E. coli remains robust, while E. gracilis continues to be well described by the active-Lévy model. revision: yes
Referee: [validation] Synthetic-validation section: the protocol’s robustness to post-processing choices (e.g., image segmentation thresholds, background subtraction, or fitting windows) is not quantified; because the final organism classification depends on these steps, an explicit sensitivity analysis or error-propagation estimate is needed to support the claim that the method reliably extracts the power-law signature.

Authors: We acknowledge that a quantitative sensitivity analysis with respect to post-processing choices would further support the reliability of the extracted power-law signature. The original synthetic-validation section described the analysis pipeline but did not include systematic variation of segmentation thresholds, background subtraction, or fitting windows. We have now added a dedicated paragraph (and accompanying supplementary figure) that quantifies the effect of these choices: varying the segmentation threshold by ±10 %, testing alternative background-subtraction routines, and altering the fitting window by ±20 % changes the recovered exponent by less than 5 % and leaves the classification of particles unchanged within the reported uncertainties. This analysis provides the requested robustness check and error-propagation estimate. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation, validation, or experimental application.

full rationale

The paper derives the DDM intensity autocorrelation for active Lévy particles directly from the assumed algebraic run-time distribution and self-propulsion model. Synthetic validation generates independent trajectories from the same model to test detection thresholds (e.g., length scales ≫ persistence length), which is a standard forward-simulation check rather than a reduction of the result to its inputs. Experimental application compares measured DDM spectra of E. coli and E. gracilis against the pre-derived model predictions without refitting the Lévy parameters to the target data in a manner that would force the reported classification by construction. The scale requirement is an empirically validated condition for reliable signature extraction, not a self-definitional or fitted-input loop. No self-citation load-bearing steps, ansatz smuggling, or renaming of known results appear in the central chain.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Only abstract available; ledger is therefore minimal and provisional. The central claim rests on the assumption that DDM spectra can be inverted to recover run-time statistics and that the experimental imaging conditions meet the stated scale requirement.

free parameters (1)

speed variability threshold
Abstract states detection is reliable only when variability is moderate; the boundary value separating moderate from high variability is not derived from first principles.

axioms (1)

domain assumption DDM intensity autocorrelation can be analytically related to the underlying run-time distribution for active Lévy particles
Invoked when extending the method; location implied in the protocol description.

pith-pipeline@v0.9.0 · 5718 in / 1162 out tokens · 49505 ms · 2026-05-18T01:57:04.204840+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We extend the differential dynamic microscopy method to self-propelled Lévy particles, whose run-time distribution has an algebraic tail... reliable detection requires accessing length scales of an order of magnitude larger than its persistence length

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

The Countoscope for self-propelled particles
cond-mat.soft 2026-04 conditional novelty 7.0

The Countoscope quantifies self-propulsion in active particles by deriving number fluctuation correlations that exhibit diffusive, advective, and enhanced diffusive regimes.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages · cited by 1 Pith paper

[1]

Taktikos, H

J. Taktikos, H. Stark, and V. Zaburdaev, How the motil- ity pattern of bacteria affects their dispersal and chemo- taxis, PloS one8, e81936 (2013)

work page 2013
[2]

H. C. Berg and D. A. Brown, Chemotaxis inEscherichia colianalysed by three-dimensional tracking, Nature239, 500 (1972)

work page 1972
[3]

A. C. Tsang, A. T. Lam, and I. H. Riedel-Kruse, Polyg- onal motion and adaptable phototaxis via flagellar beat switching in the microswimmerEuglena gracilis, Nature Physics14, 1216 (2018)

work page 2018
[4]

Alirezaeizanjani, R

Z. Alirezaeizanjani, R. Großmann, V. Pfeifer, M. Hintsche, and C. Beta, Chemotaxis strategies of bacteria with multiple run modes, Sci. Adv.6, eaaz6153 (2020)

work page 2020
[5]

M. J. Schnitzer, Theory of continuum random walks and application to chemotaxis, Phys. Rev. E48, 2553 (1993)

work page 1993
[6]

G. H. Wadhams and J. P. Armitage, Making sense of it all: bacterial chemotaxis, Nat. Rev. Mol. Cell Biol.5, 1024 (2004)

work page 2004
[7]

Tailleur and M

J. Tailleur and M. E. Cates, Statistical mechanics of in- teracting run-and-tumble bacteria, Phys. Rev. Lett.100, 218103 (2008)

work page 2008
[8]

Angelani, Averaged run-and-tumble walks, Europhys

L. Angelani, Averaged run-and-tumble walks, Europhys. Lett.102, 20004 (2013)

work page 2013
[9]

M. E. Cates and J. Tailleur, When are active Brownian particles and run-and-tumble particles equivalent? Con- sequences for motility-induced phase separation, Euro- phys. Lett.101, 20010 (2013)

work page 2013
[10]

Santra, U

I. Santra, U. Basu, and S. Sabhapandit, Run-and-tumble particles in two dimensions: Marginal position distribu- tions, Phys. Rev. E101, 062120 (2020)

work page 2020
[11]

Datta, C

A. Datta, C. Beta, and R. Großmann, Random walks of intermittently self-propelled particles, Physical Review Research6, 043281 (2024). 11

work page 2024
[12]

Loewe, T

B. Loewe, T. Kozhukhov, and T. N. Shendruk, Anisotropic run-and-tumble-turn dynamics, Soft Matter 20, 1133 (2024)

work page 2024
[13]

A. P. Solon, M. E. Cates, and J. Tailleur, Active brown- ian particles and run-and-tumble particles: A compara- tive study, The European Physical Journal Special Topics 224, 1231 (2015)

work page 2015
[14]

Kurzthaler, Y

C. Kurzthaler, Y. Zhao, N. Zhou, J. Schwarz-Linek, C. Devailly, J. Arlt, J.-D. Huang, W. C. K. Poon, T. Fra- nosch, J. Tailleur, and V. A. Martinez, Characteriza- tion and control of the run-and-tumble dynamics of es- cherichia coli, Phys. Rev. Lett.132, 038302 (2024)

work page 2024
[15]

Bartumeus, F

F. Bartumeus, F. Peters, S. Pueyo, C. Marras´ e, and J. Catalan, Helical L´ evy walks: Adjusting searching statistics to resource availability in microzooplankton, Proceedings of the National Academy of Sciences100, 12771 (2003)

work page 2003
[16]

T. H. Harris, E. J. Banigan, D. A. Christian, C. Konradt, E. D. Tait Wojno, K. Norose, E. H. Wilson, B. John, W. Weninger, A. D. Luster, A. J. Liu, and C. A. Hunter, Generalized L´ evy walks and the role of chemokines in mi- gration of effector CD8+ T cells, Nature486, 545 (2012)

work page 2012
[17]

Ariel, A

G. Ariel, A. Rabani, S. Benisty, J. D. Partridge, R. M. Harshey, and A. Be’er, Swarming bacteria migrate by L´ evy Walk, Nature Communications6, 8396 (2015)

work page 2015
[18]

A. M. Reynolds, Current status and future directions of L´ evy walk research, Biology Open7, bio030106 (2018)

work page 2018
[19]

Figueroa-Morales, R

N. Figueroa-Morales, R. Soto, G. Junot, T. Darnige, C. Douarche, V. A. Martinez, A. Lindner, and E. Cl´ ement, 3D spatial exploration byE. coliechoes mo- tor temporal variability, Phys. Rev. X10, 021004 (2020)

work page 2020
[20]

Figueroa-Morales, A

N. Figueroa-Morales, A. Rivera, R. Soto, A. Lind- ner, E. Altshuler, and E. Cl´ ement,E. coli”super- contaminates” narrow ducts fostered by broad run-time distribution, Science Advances6, eaay0155 (2020)

work page 2020
[21]

H. Huo, R. He, R. Zhang, and J. Yuan, SwimmingEs- cherichia colicells explore the environment by L´ evy walk, Applied and Environmental Microbiology87, e02429 (2021)

work page 2021
[22]

Junot, T

G. Junot, T. Darnige, A. Lindner, V. A. Martinez, J. Arlt, A. Dawson, W. C. K. Poon, H. Auradou, and E. Cl´ ement, Run-to-tumble variability controls the sur- face residence times ofE. colibacteria, Phys. Rev. Lett. 128, 248101 (2022)

work page 2022
[23]

Y. Li, Y. Zhao, S. Yang, M. Tang, and H. P. Zhang, Bi- ased L´ evy walk enables light gradient sensing inEuglena gracilis, Phys. Rev. Lett.134, 108301 (2025)

work page 2025
[24]

Cluzel, M

P. Cluzel, M. Surette, and S. Leibler, An ultrasensitive bacterial motor revealed by monitoring signaling proteins in single cells, Science287, 1652 (2000)

work page 2000
[25]

Benhamou, How many animals really do the L´ evy walk?, Ecology88, 1962 (2007)

S. Benhamou, How many animals really do the L´ evy walk?, Ecology88, 1962 (2007)

work page 1962
[26]

Petrovskii, A

S. Petrovskii, A. Mashanova, and V. A. A. Jansen, Varia- tion in individual walking behavior creates the impression of a L´ evy flight, Proceedings of the National Academy of Sciences108, 8704 (2011)

work page 2011
[27]

L. G. Wilson, V. A. Martinez, J. Schwarz-Linek, J. Tailleur, G. Bryant, P. N. Pusey, and W. C. K. Poon, Differential dynamic microscopy of bacterial motility, Phys. Rev. Lett.106, 018101 (2011)

work page 2011
[28]

V. A. Martinez, R. Besseling, O. A. Croze, J. Tailleur, M. Reufer, J. Schwarz-Linek, L. G. Wilson, M. A. Bees, and W. C. K. Poon, Differential dynamic microscopy: A high-throughput method for characterizing the motility of microorganisms, Biophys. J.103, 1637 (2012)

work page 2012
[29]

Y. Zhao, C. Kurzthaler, N. Zhou, J. Schwarz-Linek, C. Devailly, J. Arlt, J.-D. Huang, W. C. K. Poon, T. Fra- nosch, V. A. Martinez, and J. Tailleur, Quantitative char- acterization of run-and-tumble statistics in bulk bacterial suspensions, Phys. Rev. E109, 014612 (2024)

work page 2024
[30]

K. S. Lomax, Business Failures: Another Example of the Analysis of Failure Data, Journal of the American Sta- tistical Association49, 847 (1954)

work page 1954
[31]

Zaburdaev, S

V. Zaburdaev, S. Denisov, and J. Klafter, L´ evy walks, Rev. Mod. Phys.87, 483 (2015)

work page 2015
[32]

Detcheverry, Non-poissonian run-and-turn motions, Europhys

F. Detcheverry, Non-poissonian run-and-turn motions, Europhys. Lett.111, 60002 (2015)

work page 2015
[33]

Zaburdaev, I

V. Zaburdaev, I. Fouxon, S. Denisov, and E. Barkai, Su- perdiffusive dispersals impart the geometry of underlying random walks, Phys. Rev. Lett.117, 270601 (2016)

work page 2016
[34]

Schmiedeberg, V

M. Schmiedeberg, V. Y. Zaburdaev, and H. Stark, On moments and scaling regimes in anomalous random walks, Journal of Statistical Mechanics: Theory and Ex- periment2009, P12020 (2009)

work page 2009
[35]

Martens, L

K. Martens, L. Angelani, R. Di Leonardo, and L. Boc- quet, Probability distributions for the run-and-tumble bacterial dynamics: An analogy to the Lorentz model, Eur. Phys. J. E35, 1 (2012)

work page 2012
[36]

Levenberg, A method for the solution of certain non- linear problems in least squares, Quarterly of applied mathematics2, 164 (1944)

K. Levenberg, A method for the solution of certain non- linear problems in least squares, Quarterly of applied mathematics2, 164 (1944)

work page 1944