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REVIEW 4 major objections 5 minor 46 references

How Spatially Modulated Activity Reshapes Active Polymer Conformations

T0 review · 4 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Sinusoidally modulated tangential activity can collapse a semiflexible polymer into a globule at low mode numbers or swell it into alternating stretched and compressed segments at higher modes, with the switch near mode 4 analytically and m

desk verdict Solid analytic expansion for patterned active polymers, but the abstract overstates the confirmation: an uncontrolled closure and a theory-simulation threshold mismatch leave the central quantitative claim unverified. read the letter →

arxiv 2512.14478 v2 pith:2EIYTJY7 submitted 2025-12-16 cond-mat.soft

Paolo Malgaretti , Emanuele Locatelli This is my paper · ORCID
classification cond-mat.soft PACS 05.40.-a82.35.Lr
keywords activepolymerstangentialactivityRousemodelsemiflexiblegyrationradiusend-to-enddistancemode-dependentconformationsPécletnumber
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to show that spatially patterned tangential activity is a control knob for polymer conformation even when the active force is weak. Using a semiflexible Rouse model expanded in small activity, it derives mode-correlation formulas that predict a sign change in the gyration-radius correction at a threshold forcing mode m*: low-mode forcing (m = 0,1) shrinks the chain into globule-like shapes, while higher modes produce alternating stretched and compressed segments and a swollen chain. The same expansion explains how a polymer can be compact in gyration radius yet extended end-to-end, because the two metrics weight modes differently. The authors support the analytic predictions with Langevin simulations, while noting that the simulations run at finite rather than perturbatively small activity and that the analytic threshold (m* = 4) differs from the simulated one (m* = 2).

What carries the argument

The machinery is a continuum Rouse model with bending rigidity, expanded in the passive-chain eigenmodes ψ_n(s) = sqrt(2/N) cos(π n s/N), with eigenvalues ζ_n ∝ n²(1+Λ_p n²) and a single-mode tangential active force f(s) = f_m cos(π m s/N). To make the expansion tractable, the paper uses a mean-field decoupling of the normalized tangent vector: the average of r_n / |∂s r| is replaced by ⟨r_n⟩ / sqrt(⟨Θ(s)⟩), and ⟨Θ(s)⟩ is then approximated by its peak value b². This yields closed perturbative formulas for ⟨r_i·r_j⟩ at orders 0, 1, and 2 in the Péclet number Pe_m = f_m b √N / (k_B T), which is the combination that identifies the weak-activity regime uniformly in chain length.

What would settle it

Run Langevin simulations of a tangential-active polymer at Pe_m ≪ 1 (for example f_a ≈ 10⁻³ k_BT/σ with N = 400, so Pe_m ≈ 0.02) and measure R_G² as a function of the forcing mode m. The analytic prediction is a sign change in the correction at m* = 4; observing the sign flip at m* = 2 at such small Péclet numbers would falsify the quantitative closure.

Watch

Extended reading notes

Core claim

The paper's central claim is that the spatial mode of tangential self-propulsion determines whether a semiflexible polymer shrinks or swells. Under weak sinusoidally modulated propulsion, the second-order correction to the gyration radius flips sign at a threshold mode number: uniform or low-mode forcing produces compact, globule-like conformations, whereas higher modes generate alternating stretched and compressed segments that make the chain globally swollen. The analytical expressions (Eqs. 30) for mode correlations, gyration radius, and end-to-end distance show that activity breaks self-similar scaling and that different size metrics respond differently, allowing conformations that are c

Load-bearing premise

The load-bearing approximation is the mean-field replacement of the average of the normalized tangent ratio by the ratio of averaged quantities (Eq. 19), together with treating ⟨Θ(s)⟩ as a constant; the paper's own simulations could not reach the perturbatively small forces and show a shifted threshold (m* = 4 analytically vs m* = 2 numerically), so the closure is plausible but uncontrolled.

Editorial extensions

If this is right

  • Weak, spatially structured activity can induce both collapse and swelling of a single polymer, making force patterning a practical alternative to changing solvent quality.
  • Because the relevant Péclet number grows as √N at fixed force, longer chains respond much more strongly to the same active forcing.
  • A chain may be compact by gyration radius but extended end-to-end, so measuring either quantity alone can miss the conformational state.
  • For ring polymers, leading-order activity corrections vanish by parity (only even modes contribute), so topology sharply constrains the linear-polymer result.
  • Non-conservative activity yields a quadratic dependence of the gyration radius on force strength, distinguishing active collapse from conservative attraction, which is linear.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the threshold m* is governed by the normalized eigenvalues z̄_i = i²(1+Λ_p i²), one could tune persistence length or polymerization degree to place a desired spatial mode on either side of the transition, effectively programming polymer shape by choosing which modes are active.
  • The analytic-versus-simulated threshold discrepancy suggests that the mean-field closure underrepresents single-mode effects; a variational or renormalized treatment might shift m* and improve quantitative agreement at finite Péclet numbers.
  • The same weak-activity expansion could be extended to multi-mode or time-modulated forcing, where interference between modes might create regimes that neither mode produces alone, such as collapse driven by two modes that individually swell the chain.
  • The parity argument for rings is directly testable: simulations of ring polymers at small Péclet should show no leading-order gyration-radius change, with higher-order corrections and swelling emerging only as the activity grows.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper studies a semiflexible active polymer driven by spatially modulated tangential forces. Using a continuum Rouse model with bending rigidity, the authors perform a small-activity expansion of the mode correlations and derive second-order expressions for the gyration radius and end-to-end distance for a sinusoidal forcing profile. The central claim is that spatially structured activity breaks the self-similar scaling of passive chains and produces a mode-dependent transition from shrinking for low forcing modes to swelling for high modes. Langevin dynamics simulations are used to support the analytical predictions. The manuscript also reports a threshold mismatch m*=4 (theory) versus m*=2 (simulation) and concedes that the simulations could not reach the small-Péclet regime required by the expansion.

Significance. If the analytical framework is correct, it provides a useful, parameter-free route to predict how patterned tangential activity reshapes polymer conformations. The paper has clear strengths: the Péclet number is identified rather than fitted, the stretching-coefficient correction D2 is derived from a fixed-contour-length constraint, the expansions are worked out in detail, and simulation data are publicly deposited. However, the quantitative validation is not currently established. The key predictions rest on an uncontrolled mean-field closure and on a constant-⟨Θ⟩ approximation, and the simulation comparison is made at Péclet numbers far outside the expansion regime. The qualitative sign trend is plausible, but the paper's stronger claim of quantitative confirmation is not supported by the evidence presented.

major comments (4)
  1. [Sec. II B, Eq. (19)] The central approximation replaces the average of a ratio, ⟨ r_n / sqrt(Σ_m r_m² (∂_s ψ_m)²) ⟩, by the ratio of averages, ⟨r_n⟩ / sqrt(Σ_m ⟨r_m²⟩ (∂_s ψ_m)²). This mean-field-type decoupling is uncontrolled and enters the derivation of every higher-order correlation, including Eqs. (30), (42), and (46). No error estimate or consistency check is provided. The discrepancy m*=4 versus m*=2 reported in Sec. IV B is exactly the kind of quantitative failure this closure could produce. The authors should either justify the closure from a small parameter, test it against the exact stationary distribution in a simplified model, or quantify its error numerically.
  2. [Appendix A 2, Eq. (A24)] In the short-persistence-length treatment, ⟨Θ(s)⟩_0 is replaced by the constant b², discarding the spatial dependence derived in Eq. (A22), including boundary-layer variation. This approximation is used to evaluate ξ^(0) and ξ^(1), which determine the sign and magnitude of R_G,2 and R_E,2. Since the central claim depends on the sign of these corrections, the sensitivity of the threshold m* to this approximation must be examined. For example, using the full expression for ⟨Θ(s)⟩_0, or keeping the next term in the κN expansion, would show whether the predicted m*=4 is robust.
  3. [Sec. IV A, Fig. 6] The claimed 'quantitative confirmation' is not supported by the data shown. The simulations use f_a=0.01 and 10 (Pe≈0.14–0.20 and Pe≈141–200), while the expansion requires Pe_l≪1. The upper panel of Fig. 6 is in a marginal regime where the second-order correction is very small, and the lower panel is far outside the expansion regime. Moreover, the theoretical curves in Figs. 6(b,c) and 8(b,c) are computed at Pe=1 and are not overlaid with the simulation data. The paper should either present a direct quantitative comparison at a Péclet number where the expansion is expected to hold, or explicitly restrict the claim to qualitative agreement of the sign trend.
  4. [Sec. IV B, end-to-end distance] The theory and simulations disagree on the leading-order sign of R_E,1 for m=3: the text states that the theory predicts a decrease of R_E for m=3, 'in contrast to simulation results.' This is attributed to higher-order corrections, but no evidence is provided that the O(Pe) prediction becomes correct in the Pe→0 limit of the discrete model. Since this is a leading-order parity-dependent prediction, it should be tested directly or the claim should be softened.
minor comments (5)
  1. [Eq. (39)] The derivation of the gyration tensor appears to contain an algebraic slip: the term −⟨r0,i r0,j⟩ is written and then dropped, but the orthogonality of the cosines makes the center-of-mass contribution vanish for different reasons. Please clarify the notation and the cancellation.
  2. [Notation] The Péclet number is denoted Pe_l, Pe_m, and Pem in different places (Eqs. (32), (A36), and Fig. 6). Please unify the notation and define the subscript.
  3. [Sec. IV A] The sentence 'in both the upper and lower sub-panels' is misleading because the two panels correspond to very different Péclet numbers; the qualitative statement should be separated from the quantitative comparison.
  4. [Sec. IV A, Figs. 6-8] The theoretical curves are shown for Pe=1, but the text sometimes reads as if they were directly comparable to the simulation data at Pe=0.141–200. Please state explicitly that the analytical plots are illustrative of the weak-activity expansion.
  5. [Sec. II D, Eq. (27)] The shift of domain from [−N/2,N/2] to [0,N] is performed without stating the corresponding boundary conditions in terms of the shifted eigenfunctions. A brief comment would help readers verify the trigonometric representation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the weak-activity expansion is self-contained and independently checked against simulation.

full rationale

The paper’s central predictions—mode correlations, gyration radius, and end-to-end distance—are obtained from a systematic expansion of the continuum Rouse model (Eqs. 1 and 15), not from quantities that already encode the target results. The mean-field replacement in Eq. (19) and the constant-Θ approximation in Eq. (A24) are uncontrolled approximations, but they are not fitted to the simulation outcomes; the predicted second-order corrections are then computed, not imposed. The stretching-coefficient correction D2 is derived from the fixed-contour-length constraint (Eqs. A40–A51), independent of the gyration-radius or end-to-end data, so it does not constitute a fitted input called a prediction. The Péclet number in Eq. (32) is identified from the order-by-order structure of the expansion, and the paper explicitly notes it was also empirically found in prior work [34]; this is a consistency check rather than a load-bearing self-citation. The prior author works [32, 34] provide motivation and a model context, but the new analytical expressions and the predicted mode-dependent transition are derived here. Moreover, the theory is not calibrated to simulations: the authors report a quantitative mismatch in the threshold (m*=4 in theory versus m*=2 in simulations) and attribute it to model idealizations. This external, unadjusted comparison is evidence against circularity. The main concerns—validity of the mean-field closure and inability to simulate the true small-Pe regime—are correctness risks, not circularity. No step in the derivation reduces by construction to its inputs, so the honest finding is no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central results rest on a perturbative expansion whose closure uses an uncontrolled mean-field approximation and a constant approximation for the mean-square tangent, plus the short-l_p limit. No free constants are fitted to simulation data; the Péclet number is identified from the expansion and the length-fixing correction D₂ is derived.

assumptions (5)
  • ad hoc to paper Mean-field decoupling: ⟨ r_n / sqrt(Σ_m r_m² (∂_s ψ_m)²) ⟩ ≈ ⟨r_n⟩ / sqrt(Σ_m ⟨r_m²⟩ (∂_s ψ_m)²)
    Introduced in Eq. (19) to close the moment equations; no controlled error estimate; underpins the entire perturbative expansion.
  • standard math Passive eigenfunctions (free-end boundary conditions) form a complete basis for the driven problem
    Used in Sec. II.A to expand the solution; standard for the linear passive operator, but the active term is nonlinear and the expansion is not proven to converge.
  • domain assumption Short persistence length limit l_p << N: k_n = πn/N, β_n → ∞, hyperbolic functions neglected
    Sec. II.D; valid for flexible polymers and consistent with simulation l_p/L=0.0075, but breaks for stiffer chains.
  • ad hoc to paper ⟨Θ(s)⟩_0 is approximated by its maximum value b², discarding spatial dependence
    Eqs. (A22)-(A24); used to evaluate the denominator; relies on κN >> 1 and ignores edge effects.
  • domain assumption Polymer contour length is fixed: corrections D₁, D₂ to the stretching coefficient enforce L = bN
    Appendix A4; physically motivated for real polymers, but the model without this constraint would allow stretching; the correction is derived, not fitted.

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Pith. "Pith review of How Spatially Modulated Activity Reshapes Active Polymer Conformations." pith.science (2026). https://pith.science/paper/2EIYTJY7

@misc{pith2026251214478,
  author       = {Pith},
  title        = {Pith review of: How Spatially Modulated Activity Reshapes Active Polymer Conformations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2EIYTJY7}},
  note         = {Machine review of arXiv:2512.14478}
}
read the original abstract

Active polymers are driven out of equilibrium by internal forces and exhibit conformational properties that differ fundamentally from those of passive chains. Here we study how spatially modulated tangential activity reshapes the conformations of semiflexible polymers. Using a continuum Rouse model with bending rigidity, we develop a systematic expansion in the limit of weak activity and derive analytical expressions for mode correlations, gyration radius, and end-to-end distance under sinusoidally varying propulsion. We show that spatially structured activity breaks self-similar scaling and induces a mode-dependent transition between polymer shrinking and swelling. Uniform or low-mode forcing produces compact, globule-like conformations, whereas higher modes generate alternating stretched and compressed segments, leading to globally swollen chains. Different polymer sizes respond differently to activity, allowing for conformations that are compact in gyration radius yet extended in end-to-end distance. Langevin dynamics simulations quantitatively confirm the theoretical predictions. Our results demonstrate that even weak, patterned activity provides a powerful mechanism to control polymer conformations far from equilibrium.

Figures

Figures reproduced from arXiv: 2512.14478 by the authors.

Figure 1
Figure 1. FIG. 1: (a) Sketch of the continuum active polymer [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Snapshots of active polymers, [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: More snapshots of active polymers, [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: (a) Ratio of the gyration radius, over the equilibrium value [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: (a) Ratio of the end-to-end square distance, over the equilibrium value [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: (a): Elongation of the polymer ( [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]

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Works this paper leans on

46 extracted references · 1 linked inside Pith

  1. [1]

    M. C. Marchetti, J.-F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Reviews of Modern Physics85, 1143 (2013)

    2013

  2. [2]

    M. E. Cates and J. Tailleur, Annu. Rev. Cond. Matt. Phys.6, 219 (2015)

    2015

  3. [3]

    Tjhung, C

    E. Tjhung, C. Nardini, and M. E. Cates, Proceedings of the National Academy of Sciences115, E1174 (2018)

    2018

  4. [4]

    Narayan, S

    V. Narayan, S. Ramaswamy, and N. Menon, Science317, 105 (2007)

    2007

  5. [5]

    H. H. Wensink, J. Dunkel, S. Heidenreich, K. Drescher, R. E. Goldstein, H. L¨ owen, and J. M. Yeomans, Proceedings of the National Academy of Sciences109, 14308 (2012)

    2012

  6. [6]

    Needleman and Z

    D. Needleman and Z. Dogic, Nature Reviews Materials2, 17048 (2017)

    2017

  7. [7]

    Vicsek, A

    T. Vicsek, A. Czir´ ok, E. Ben-Jacob, I. Cohen, and O. Shochet, Physical Review Letters75, 1226 (1995)

    1995

  8. [8]

    Howard, Mechanics of Motor Proteins and the Cytoskeleton (Sinauer Associates, Sunderland, MA, 2001)

    J. Howard, Mechanics of Motor Proteins and the Cytoskeleton (Sinauer Associates, Sunderland, MA, 2001)

    2001

Show all 46 references
  1. [9]

    Alberts, D

    B. Alberts, D. Bray, K. Hopkin, A. D. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, Essential cell biology (Garland Science, 2015)

    2015

  2. [10]

    Zidovska, D

    A. Zidovska, D. A. Weitz, and T. J. Mitchison, Proceedings of the National Academy of Sciences110, 15555 (2013)

    2013

  3. [11]

    Eshghi, A

    I. Eshghi, A. Zidovska, and A. Y. Grosberg, Physical Review Letters131, 048401 (2023)

    2023

  4. [12]

    Mahajan, W

    A. Mahajan, W. Yan, A. Zidovska, D. Saintillan, and M. J. Shelley, Physical Review X12, 041033 (2022)

    2022

  5. [13]

    Goychuk, D

    A. Goychuk, D. Kannan, A. K. Chakraborty, and M. Kardar, Proceedings of the National Academy of Sciences120, e2221726120 (2023)

    2023

  6. [14]

    Remini, M

    L. Remini, M. Segers, J. Palmeri, J.-C. Walter, A. Parmeggiani, and E. Carlon, Physical Review E109, 024408 (2024)

    2024

  7. [15]

    G. J. Narlikar, R. Sundaramoorthy, and T. Owen-Hughes, Cell154, 490 (2013)

    2013

  8. [16]

    Lauga and T

    E. Lauga and T. R. Powers, Reports on Progress in Physics72, 096601 (2009)

    2009

  9. [17]

    M. K. Faluweki, J. Cammann, M. G. Mazza, and L. Goehring, Physical Review Letters131, 158303 (2023)

    2023

  10. [18]

    Kurjahn, L

    M. Kurjahn, L. Abbaspour, F. Papenfuß, P. Bittihn, R. Golestanian, B. Mahault, and S. Karpitschka, Nature Communi- cations15, 9122 (2024)

    2024

  11. [19]

    Rosko, R

    J. Rosko, R. N. Poon, K. Cremin, E. Locatelli, M. Coates, S. J. Duxbury, K. Randall, K. Croft, C. Valeriani, M. Polin, et al., Elife13, RP100768 (2025)

    2025

  12. [20]

    Deblais, A

    A. Deblais, A. C. Maggs, D. Bonn, and S. Woutersen, Phys. Rev. Lett.124, 208006 (2020)

    2020

  13. [21]

    Deblais, K

    A. Deblais, K. Prathyusha, R. Sinaasappel, H. Tuazon, I. Tiwari, V. P. Patil, and M. S. Bhamla, Soft Matter19, 7057 (2023)

    2023

  14. [22]

    Sinaasappel, M

    R. Sinaasappel, M. Fazelzadeh, T. Hooijschuur, Q. Di, S. Jabbari-Farouji, and A. Deblais, Physical Review Letters134, 128303 (2025)

    2025

  15. [23]

    Dreyfus, J

    R. Dreyfus, J. Baudry, M. L. Roper, M. Fermigier, H. A. Stone, and J. Bibette, Nature437, 862 (2005)

    2005

  16. [24]

    Chelakkot, A

    R. Chelakkot, A. Gopinath, L. Mahadevan, and M. F. Hagan, Journal of The Royal Society Interface11, 20130884 (2014)

    2014

  17. [25]

    H. R. Vutukuri, B. Bet, R. Van Roij, M. Dijkstra, and W. T. Huck, Scientific reports7, 16758 (2017)

    2017

  18. [26]

    Nishiguchi, J

    D. Nishiguchi, J. Iwasawa, H.-R. Jiang, and M. Sano, New Journal of Physics20, 015002 (2018), ISSN 1367-2630

    2018

  19. [27]

    Martinet, Y

    Q. Martinet, Y. Li, A. Aubret, E. Hannezo, and J. Palacci, Physical Review X15, 041017 (2025)

    2025

  20. [28]

    Kumar, A

    M. Kumar, A. Murali, A. G. Subramaniam, R. Singh, and S. Thutupalli, Nature Communications15, 4903 (2024)

    2024

  21. [29]

    A. G. Subramaniam, M. Kumar, S. Thutupalli, and R. Singh, New Journal of Physics26, 083009 (2024). 14

    2024

  22. [30]

    Ozkan-Aydin, D

    Y. Ozkan-Aydin, D. I. Goldman, and M. S. Bhamla, Proceedings of the National Academy of Sciences118, e2010542118 (2021)

    2021

  23. [31]

    Becker, C

    K. Becker, C. Teeple, N. Charles, Y. Jung, D. Baum, J. C. Weaver, L. Mahadevan, and R. Wood, Proceedings of the National Academy of Sciences119, e2209819119 (2022)

    2022

  24. [32]

    Bianco, E

    V. Bianco, E. Locatelli, and P. Malgaretti, Phys. Rev. Lett.121, 217802 (2018)

    2018

  25. [33]

    R. E. Isele-Holder, J. Elgeti, and G. Gompper, Soft Matter11, 7181 (2015), ISSN 1744-683X, URLhttp://xlink.rsc. org/?DOI=C5SM01683E

    2015

  26. [34]

    Malgaretti, E

    P. Malgaretti, E. Locatelli, and C. Valeriani, Molecular Physics p. e2384462 (2024)

    2024

  27. [35]

    Jaiswal, P

    S. Jaiswal, P. Maity, S. Thakur, and M. Ripoll, Arxiv p. 2503.18099 (2025)

    2025 arXiv

  28. [36]

    Harnau, R

    L. Harnau, R. G. Winkler, and P. Reineker, The Journal of chemical physics102, 7750 (1995)

    1995

  29. [37]

    Eisenstecken, G

    T. Eisenstecken, G. Gompper, and R. G. Winkler, Polymers8(2016)

    2016

  30. [38]

    A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M. Brown, P. S. Crozier, P. J. In’t Veld, A. Kohlmeyer, S. G. Moore, T. D. Nguyen, et al., Computer physics communications271, 108171 (2022)

    2022

  31. [39]

    Fazelzadeh, E

    M. Fazelzadeh, E. Irani, Z. Mokhtari, and S. Jabbari-Farouji, Phys. Rev. E108, 024606 (2023)

    2023

  32. [40]

    Locatelli, V

    E. Locatelli, V. Bianco, and P. Malgaretti, Phys. Rev. Lett.126, 097801 (2021)

    2021

  33. [41]

    Breoni, E

    D. Breoni, E. Locatelli, and L. Tubiana, Macromolecules58, 10677 (2025)

    2025

  34. [42]

    J. P. Miranda-Lopez, E. Locatelli, C. Micheletti, D. Levis, and C. Valeriani, Journal of Chemical Theory and Computation 21, 11910 (2025). Appendix A: Details on the analytical expansions

    2025

  35. [43]

    (22) analytically consists of a systematic expansion of all modes at small values off a

    Analytical Expansions I: small active forces Our strategy for tackling Eq. (22) analytically consists of a systematic expansion of all modes at small values off a. In order to calculate⟨r i(t)·r j⟩at the different orders inf a we need to calculate⟨r i(t)·η j⟩. We will employ t...

  36. [44]

    Analytical Expansion II: Short persistence length In order to calculate the last expression we assume that the persistence length is small,l p ≪N. As reported in the main text, in this regime we have thatβ 2 n =k 2 n + 1/l2 p ≫N: the hyperbolic functions are of order unity onl...

  37. [45]

    Is interesting to note that the P´ eclet number in Eq

    Single force mode and relevant P´ eclet number In the case of a single Fourier mode of the active force, by comparing the values of the correlation at zeroth order to those at first order we obtain that the latter are much smaller then the former when Pel = flb √ N kBT ≪1,(A36...

  38. [46]

    Since for real polymers this is rarely the case, we have computed the corrections that should be added to the ”stretching” parameter D(see Eq

    Polymer length & stretching coefficient In the continuum model the total length of the polymer is not fixed and the polymer can be stretched. Since for real polymers this is rarely the case, we have computed the corrections that should be added to the ”stretching” parameter D(...

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