Pith. sign in

REVIEW 2 major objections 6 minor 81 references

Effects of knotting on the collapse of active ring polymers

T0 review · 2 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Knot topology shifts the collapse point of tangentially active ring polymers: twist knots collapse earlier, and complex twist knots lose the actively stretched regime entirely, while torus knots become more resistant as their crossing…

desk verdict New torus-vs-twist collapse result with a solid torus branch; the twist 'vanishing stretched regime' rests on an excluded point and extrapolation. read the letter →

arxiv 2507.08391 v2 pith:6SDETGKY submitted 2025-07-11 cond-mat.soft cond-mat.stat-mech

Davide Breoni , Emanuele Locatelli , Luca Tubiana This is my paper · ORCID
classification cond-mat.softcond-mat.stat-mech PACS 61.25.H82.35.Lr
keywords activepolymersringknottopologycollapsetransitiontangentialactivitytorusknotstwistKremer-Grestmodel
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 asks whether the knot tied in a ring polymer changes how the polymer behaves when its monomers actively push themselves along the chain. Simulating tangentially active rings with different knots, it finds that knot family and knot complexity shift the polymer size at which the active stretched state collapses: torus knots collapse later as their minimal crossing number grows, while twist knots collapse earlier, and sufficiently complex twist knots lose the stretched regime altogether. The proposed mechanism is geometric: stretched torus knots relax into ordered double-helix conformations whose strands move in the same direction, suppressing collisions, whereas twist knots carry noose-like constraints that seed collisions and deadlock-driven collapse. If correct, topology becomes a tunable handle on active polymer size and phase behavior.

What carries the argument

The load-bearing objects are the collapse point $N_C$ and two structural observables: the bond correlation function $\beta(\delta) = \langle \sum_{i,j} \mathbf{t}_i \cdot \mathbf{t}_j / N \rangle$, whose argmin gives the typical loop size, and the density $\rho_{3D}$ of close oppositely oriented non-neighboring bonds, which measures collision-prone strand contacts. The mechanism is that stretched torus knots arrange as intertwined double helices with aligned bonds ($\beta$ minimum near $\delta/N = 1/4$), keeping $\rho_{3D}$ near zero, while twist knots lack such order and accumulate noose-like constraints that raise $\rho_{3D}$ and seed deadlocks. Knot complexity enters through the ideal knot length/diameter ratio $p$, which for torus knots grows linearly with $\mathrm{MCN}$.

What would settle it

Simulate tangentially active twist knots with ideal length/thickness ratio $p \gtrsim 40$ (minimal crossing number $\gtrsim 10$) with sufficient statistics and extract $N_C$; if $N_C$ does not keep decreasing below the unknot threshold and instead flattens or rises, the claimed disappearance of the stretched regime for complex twist knots is refuted.

Watch

Extended reading notes

Core claim

The central discovery is a family-dependent collapse transition in tangentially active ring polymers. For unknotted rings there is a known activity-induced transition from a stretched to a collapsed state at a polymerization degree $N_C$. Extending this to knotted rings, the paper shows $N_C$ scales linearly with the ideal knot length/diameter ratio $p$ (equivalently minimal crossing number $\mathrm{MCN}$) for double-helix torus knots, whereas for twist knots $N_C$ decreases with $p$; extrapolation suggests twist knots with $\mathrm{MCN} \gtrsim 10$ no longer support an extended active phase. The cause is the shape of the stretched state: torus knots form regular double helices in which non-neighboring bonds are aligned, keeping the density $\rho_{3D}$ of oppositely oriented close bonds near zero, while twist knots display noose-like regions that create collisions and deadlocks, triggering collapse.

Load-bearing premise

The twist-knot branch of the central claim rests on a decreasing fit of collapse point versus knot complexity from which the most complex twist knot studied, $9_2$, was excluded, and on an extrapolation to complexities beyond the simulated range; if including $9_2$ or higher-complexity data overturns this downward trend, the claimed torus-versus-twist dichotomy would be substantially weakened.

Editorial extensions

If this is right

  • Knot complexity becomes a design parameter: torus knots with many crossings yield active rings that stay extended at arbitrarily large lengths, while complex twist knots collapse as soon as they are long enough to tie.
  • The collapse point of a torus-knotted active ring can be predicted from the ideal knot's length/thickness ratio $p$, connecting a geometric knot invariant to a dynamical phase boundary.
  • Activity should bias the knot spectrum of a population of randomly knotted active rings toward torus knots, since complex twist knots cannot form or persist in the extended state.
  • The double-helix stretched conformation suppresses inter-strand collisions, so observables like $\rho_{3D}$ in the stretched state could serve as an early indicator of collapse onset.

Reading between the lines

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

  • If the twist-knot trend holds experimentally, knot type could act as a reporter of the active phase: measuring the knot spectrum of active DNA rings might reveal whether the population was in the stretched or collapsed regime.
  • Hydrodynamic interactions are neglected here; a testable extension is to repeat the simulations with explicit solvent and check whether the linear growth of $N_C$ for torus knots survives.
  • The mechanism suggests a route to knot-selective sorting: choosing activity levels such that twist knots collapse while torus knots remain extended could separate knots by family in a microfluidic device.
  • The collapsed state resembles metastable tight knots in semiflexible polymers; comparing collapse kinetics of active rings with passive tight knots could isolate the role of activity in stabilizing the collapsed phase.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The manuscript reports molecular dynamics simulations of tangentially active ring polymers with various knot topologies (unknot, double-helix torus knots, and twist knots), for polymerization degrees N from 20 to 1024 at Péclet number Pe=10. The authors extract a collapse transition point N_C from the probability that the gyration radius falls below 0.9 times the passive value, and they find that torus knots collapse at larger N than twist knots, that N_C increases approximately linearly with the ideal length/diameter ratio p for torus knots while decreasing for twist knots, and that the stretched regime for twist knots is predicted to disappear for p≳40. The proposed mechanism is that stretched torus knots adopt ordered double-helical conformations with aligned non-neighboring bonds that suppress bead collisions, whereas twist knots contain noose-like constraints that promote collisions and collapse.

Significance. If the main claims hold, the paper offers a clear and potentially useful demonstration that knot topology can control the activity-driven swelling-collapse transition of ring polymers, and it proposes a mechanistic picture based on bond alignment and intra-molecular collision density. The systematic coverage of many topologies, the wide range of polymerization degrees, and the explicit measurements of bond correlations and collision density are strengths. The paper is likely to interest researchers in active matter, polymer physics, and knot theory. However, the most novel quantitative claim for twist knots rests on a limited dataset and an extrapolation beyond the simulated range, so the significance is conditional on strengthening that branch.

major comments (2)
  1. [Results: The collapse transition (Fig. 3)] The decreasing trend of N_C−p with p for twist knots, which underlies the abstract's claim that the stretched regime is 'eventually canceling' and the discussion's speculation about activity biasing the knot spectrum, is not adequately supported. The text states that the linear fit for twist knots excludes the most complex simulated twist knot, 9_2, 'because of the difficulty of measuring its N_C,' and no twist knot with p≳40 or MCN≥10 is simulated; the predicted disappearance at p≳40 is an extrapolation. If the true N_C of 9_2 lies above the fitted line, or if additional higher-complexity twist knots flatten or reverse the trend, the claimed torus-versus-twist dichotomy in the collapse-point scaling would be substantially weakened. Please report the fit including 9_2 with its uncertainty, simulate at least one or two additional twist knots beyond 9_2, or explicitly demote the 'eventually canceling' statement to a conjecture.
  2. [SI: Determination of the collapsing point] The collapse point N_C is extracted from P(N) = P(R_g < 0.9 <R_g^0(N)>), where the factor 0.9 is arbitrary and is introduced to keep P close to zero at small N. Since every quantitative result in the paper is a statement about N_C and its dependence on p, the authors should demonstrate that the family ordering and the linear increase/decrease of N_C are insensitive to the chosen threshold, for example by repeating the analysis with thresholds in a range such as 0.85–0.95. Without this sensitivity check, the linear fits in Fig. 3b could reflect the threshold choice rather than a physical trend.
minor comments (6)
  1. [Methods, Eq. (1)] The WCA cutoff is written as r_c = 6√2 σ, which does not match the standard cutoff 2^(1/6) σ for the WCA potential; please verify the intended formula and correct the typography.
  2. [Methods, Langevin dynamics] The damping time is written as 'τ γ = .3τ', which is unclear; presumably this means τ_γ = 3τ, but the notation should be clarified.
  3. [Results and Fig. 2 caption] The quantity N_T is used to delimit the collapse threshold range (135≲N_T≲145) but is never formally defined; please define it and state how it is determined.
  4. [Abstract and Fig. 3] The abstract states that the collapse point 'grows linearly with the minimum crossing number,' while the analysis in Fig. 3 plots N_C against the ideal length/diameter ratio p; please state explicitly how p maps to the minimum crossing number for the simulated knots.
  5. [Results: Bead collisions and loop formation] The volume V_P = N π/4 σ^3 is described as the volume of a tube of radius 2σ and length Nσ, but such a tube has volume 4π N σ^3; this inconsistency affects the reported absolute values of ρ_3D (e.g., the asymptotic value ~0.3σ^{-3}), even though relative comparisons across topologies may be unaffected.
  6. [Results: The collapse transition (Fig. 3b)] The linear fits in Fig. 3b are described in the caption but no fit parameters, uncertainties, or goodness-of-fit measures are reported; please include them in the caption or in the SI.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the central collapse-point comparison is measured from independent simulations; the only self-citation is interpretive and not load-bearing.

full rationale

The paper's central claims are derived from simulations, not from self-referential definitions. The collapse point N_C is an empirical quantity obtained from each topology's gyration-radius distribution (SI: 'To determine the collapse point N_C ... we use a special cumulative distribution of R_g ... allowing us to extract N_C by fitting this curve to a sigmoid'). The complexity variable p is taken from external ideal-knot literature ('the parameter p = L/d, the ratio between the length and thickness of an ideal knot'), not fitted here or defined in terms of N_C. The torus-versus-twist split is directly visible in measured R_g(N) curves (Fig. 1b,c) and in the rho_3D collision statistics, which are independent observables. The linear fit of N_C versus p (Fig. 3) is a summary of measured data, not a construction forcing the result; a fit cannot be circular unless its input and output coincide by definition, and here the output (collapse point) is measured separately from the input (ideal-knot p). The only self-citation, Ref. 63 (Locatelli et al.), supplies the prior unknot collapse result and the 20-bead scale; the unknot transition is reproduced in this paper's own simulations, and the ~20-bead increment is taken from the present linear fit, so the citation is interpretive rather than load-bearing. The paper's own limitation statements, including that the linear fit in Fig. 3b 'ignore[s]' the 9_2 twist knot 'because of the difficulty of measuring its N_C' and that the predicted disappearance at p >= 40 is an extrapolation beyond simulated knots, are robustness/correctness concerns about the twist-knot branch rather than circularity: N_C remains an independently measured quantity. Thus no step in the derivation reduces by construction or by self-citation to its own input; score 2 reflects only the minor, non-load-bearing self-reference.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard polymer simulation methodology, literature values of ideal knot length, and several hand-set parameters (Pe, potential strengths, collapse threshold). No new physical entities are introduced.

free parameters (7)
  • Péclet number Pe = 10
    Activity strength; set to a single value by the authors. The collapse and its topology dependence are reported only at Pe=10.
  • WCA energy epsilon = 300 k_BT
    Repulsion strength chosen to prevent bond crossing/topology changes; hand-set.
  • FENE maximum bond length R0 = 1.05 sigma
    Small value chosen to make FENE very stiff and preserve topology.
  • FENE stiffness K = 30 epsilon/sigma^2
    Standard Kremer-Grest value; hand-set.
  • Bending stiffness kappa = 1 k_BT
    Bending potential strength; hand-set, gives a flexible chain.
  • Collapse probability threshold = 0.9
    In P(N) = P(R_g < 0.9 <R_g0>); the factor is chosen ad hoc to make P small at low N and affects N_C values.
  • Collision pair cutoffs = distance < 2 sigma; |i-j| > 3
    Hand-set definitions for which bond pairs count as close in rho_3D; changing them changes collision densities.
assumptions (6)
  • domain assumption The Kremer-Grest model with WCA excluded volume and FENE bonds accurately represents flexible polymer behavior in good solvent.
    Standard polymer simulation model; used to produce the passive scaling R_g ~ N^0.588.
  • domain assumption Ideal knot length/diameter ratios p from Refs 64,65,72,73 are valid descriptors of knot complexity for the simulated topologies.
    p is the independent variable in Fig. 3; if p values are inaccurate, the family trends would shift.
  • domain assumption The FENE/WCA parameters (epsilon=300 kBT, R0=1.05 sigma) prevent changes of knot topology during the active runs.
    Stated in Methods; no post-run knot verification is reported.
  • domain assumption Simulation time of 1e6 tau and 30 independent realizations are sufficient to estimate steady-state N_C despite glassy dynamics.
    The authors acknowledge diverging autocorrelation times and use multiple realizations to mitigate.
  • ad hoc to paper The cumulative probability P(R_g < 0.9 <R_g0>) fitted to a sigmoid yields a meaningful collapse point N_C.
    The 0.9 threshold and sigmoid form are chosen by the authors; different choices would change N_C.
  • ad hoc to paper The density rho_3D of close, oppositely oriented bond pairs is a valid proxy for collision probability and deadlock formation.
    The causal link between rho_3D and collapse is assumed, not directly tested.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Effects of knotting on the collapse of active ring polymers." pith.science (2026). https://pith.science/paper/6SDETGKY

@misc{pith2026250708391,
  author       = {Pith},
  title        = {Pith review of: Effects of knotting on the collapse of active ring polymers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6SDETGKY}},
  note         = {Machine review of arXiv:2507.08391}
}
read the original abstract

We use numerical simulations to study tangentially active flexible ring polymers with different knot topologies. Simple, unknotted active rings display a transition from an extended phase to a collapsed one upon increasing the degree of polymerization. We find that topology has a significant effect on the polymer size at which the collapse takes place, with twist knots collapsing earlier than torus knots. Increasing knot complexity further accentuates this difference, as the collapse point of torus knots grows linearly with the minimum crossing number of the knot while that of twist knots shrinks, eventually canceling the actively stretched regime altogether. This behavior is a consequence of the ordered configuration of torus knots in their stretched active state, featuring an effective alignment for non-neighboring bonds which increases with the minimal crossing number. Twist knots do not feature ordered configurations or bond alignment, increasing the likelihood of collisions, leading to collapse. These results show that topology yields a degree of control on the properties of active ring polymers, and can be used to tune them. At the same time, they suggest that activity might introduce a bias for torus knots, as complex twist knots cannot be formed in extended active polymers.

Figures

Figures reproduced from arXiv: 2507.08391 by the authors.

Figure 1
Figure 1. Gyration radius Rg of active (—) and passive (- -) ring polymers with various topologies: unknot 01 (a) double-helix torus T2 (b), and twist (c) knots as a function of polymerization N. The lines at the bottom represent the position of the collapsing point NC, with the black longer dash-dot line representing NC for unknots (estimation details in SI). The insets in (a) are simulation snapshots of unknots with differe… view at source ↗
Figure 2
Figure 2. Bond correlation function β(δ) of active ring polymers with various topologies: unknot 01, torus T2 (a,b,c) and twist (d,e,f) for different values of N. The red line remarks the collapse transition, which happens before 135 ≲ NT ≲ 145 for all twist knots and after NT for all T2 ones. Within this phenomenology, trefoils and unknots behave as torus knots. 9 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. (a) Collapsing point NC for double-helix torus (red upper-facing triangles) and twist knots (blue downward-facing triangles) as a function of the ideal length/diameter ratio p. The trefoil knot is shown with both torus and twist symbols. The dotted black line represents the collapsing threshold for the unknot, while the dashed lines are meant to underline the behavior of NC for the different families. The green area… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: (a,b) Number density of close bonds oppositely oriented [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: Probability P ≡ P(Rg < 0.9⟨R0 g ⟩) for double-helix torus (a), unknot and triple￾helix torus (b), twist knots as a function of N. The shaded area in (a) delimits the position and uncertainty of the collapsing point NC ± η. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]
Figure 6
Figure 6. Figure 6: Collapsed conformation of active ring polymers with activity [PITH_FULL_IMAGE:figures/full_fig_p027_6.png]
Figure 7
Figure 7. Figure 7: Torsional order parameter UT /N of active (—) and passive (- -) ring polymers with various topologies: unknot 01 (a), double-helix torus T2 (b), and twist (c) knot as a function of N. help of the angular momentum density l, defined as: l ≡ * m N [PITH_FULL_IMAGE:figur…
Figure 8
Figure 8. Figure 8: Angular momentum density l of active (—) and passive (- -) ring polymers with various topologies: unknot 01 (a), double-helix torus T2 (b), and twist (c) knot as a function of N. jects (short and wide) and prolate objects (thin and long). Both quantities are computed s…
Figure 9
Figure 9. Figure 9: Asphericity A of passive (a,b) and active (c,d) ring polymers with various topolo￾gies: unknot 01, double-helix torus T2 (a,c), and twist (b,d) knots as a function of N. The prolateness P is measured as P ≡ (2a − b − c)(2b − a − c)(2c − a − b) 2(a 2 + b 2 + c 2 − ab − …
Figure 10
Figure 10. Figure 10: Prolateness P of passive (a,b) and active (c,d) ring polymers with various topolo￾gies: unknot 01, double-helix torus T2 (a,c), and twist (b,d) knots as a function of N. Gyration radius and bond correlation function of passive polymers In [PITH_FULL_IMAGE:figures/ful…
Figure 11
Figure 11. Figure 11: Rg of passive ring polymers with various topologies as a function of polymerization N (a) and rescaled to take into account knot complexity (b) [PITH_FULL_IMAGE:figures/full_fig_p032_11.png]
Figure 12
Figure 12. Figure 12: Bond correlation function β(δ) of passive ring polymers with various topologies: unknot 01, torus T2 (a,b,c) and twist (d,e,f) for different values of N [PITH_FULL_IMAGE:figures/full_fig_p033_12.png]
Figure 13
Figure 13. Figure 13: (a) Gyration radius Rg of passive (- -) and active(—) T3 knots, as a function of the polymer length N. (b) Collapse point NC of T2 (red upward triangles), T3 (green circles) and twist knots (blue downward triangles) as a function of the ideal length/diameter ratio p 3…
Figure 14
Figure 14. Figure 14: (a) Bond correlation function β(δ) of active ring T3 polymers for N = 128. (b) Number density of close bonds oppositely oriented ρ3D of active ring T3 polymers as a function of the polymer length N. The inset is a simulation snapshot of an 819 knot with Pe = 10, N = 6…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

81 extracted references · 60 canonical work pages

  1. [1]

    Tubiana, L. et al. Topology in soft and biological matter. Physics Reports 2024, 1075, 1--137, DOI: doi:10.1016/j.physrep.2024.04.002

    doi:10.1016/j.physrep.2024.04.002 2024

  2. [2]

    M.; Grayson, S

    Haque, F. M.; Grayson, S. M. The synthesis, properties and potential applications of cyclic polymers. Nature Chemistry 2020, 12, 433--444, DOI: doi:10.1038/s41557-020-0440-5

    work page doi:10.1038/s41557-020-0440-5 2020

  3. [3]

    Topological and physical links in soft matter systems

    Orlandini, E.; Micheletti, C. Topological and physical links in soft matter systems. Journal of Physics: Condensed Matter 2021, 34, 013002

    2021

  4. [4]

    Soft Matter 2016, 12, 9485--9500

    On the tree-like structure of rings in dense solutions. Soft Matter 2016, 12, 9485--9500

    work page 2016

  5. [5]

    Topological gelation of reconnecting polymers

    Bonato, A.; Marenduzzo, D.; Michieletto, D.; Orlandini, E. Topological gelation of reconnecting polymers. Proceedings of the National Academy of Sciences 2022, 119, e2207728119

    work page 2022

  6. [6]

    F.; Hertzog, J

    Hart, L. F.; Hertzog, J. E.; Rauscher, P. M.; Rawe, B. W.; Tranquilli, M. M.; Rowan, S. J. Material properties and applications of mechanically interlocked polymers. Nature Reviews Materials 2021, 6, 508--530, DOI: doi:10.1038/s41578-021-00278-z

    work page doi:10.1038/s41578-021-00278-z 2021

  7. [7]

    Tying a molecular knot with optical tweezers

    Arai, Y.; Yasuda, R.; Akashi, K.-i.; Harada, Y.; Miyata, H.; Kinosita, K.; Itoh, H. Tying a molecular knot with optical tweezers. Nature 1999, 399, 446--448, DOI: doi:10.1038/20894

    work page doi:10.1038/20894 1999

  8. [8]

    J.; Stasiak, A.; Millett, K

    Jamroz, M.; Niemyska, W.; Rawdon, E. J.; Stasiak, A.; Millett, K. C.; Sułkowski, P.; Sulkowska, J. I. KnotProt : a database of proteins with knots and slipknots. Nucleic Acids Research 2015, 43, D306--D314, DOI: doi:10.1093/nar/gku1059

    work page doi:10.1093/nar/gku1059 2015

Show all 81 references
  1. [9]

    Knots are Generic Stable Phases in Semiflexible Polymers

    Majumder, S.; Marenz, M.; Paul, S.; Janke, W. Knots are Generic Stable Phases in Semiflexible Polymers . Macromolecules 2021, 54, 5321--5334, DOI: doi:10.1021/acs.macromol.0c02584

    2021 doi

  2. [10]

    Topological spectra and entropy of chromatin loop networks

    Bonato, A.; Chiang, M.; Corbett, D.; Kitaev, S.; Marenduzzo, D.; Morozov, A.; Orlandini, E. Topological spectra and entropy of chromatin loop networks. Physical Review Letters 2024, 132, 248403

    2024

  3. [11]

    Harju, J.; Broedersz, C. P. Physical models of bacterial chromosomes. Molecular Microbiology 2025, 123, 143--153, DOI: doi:10.1111/mmi.15257

    2025 doi

  4. [12]

    Ramakrishnan, S.; Chen, Z.; Fosado, Y. A. G.; Tubiana, L.; Vanderlinden, W.; Savill, N. J.; Schnaufer, A.; Michieletto, D. Single- Molecule Morphology of Topologically Digested Olympic Networks . PRX Life 2024, 2, 013009, DOI: doi:10.1103/PRXLife.2.013009

    2024 doi

  5. [13]

    W.; Roca, J

    Arsuaga, J.; Vazquez, M.; McGuirk, P.; Trigueros, S.; Sumners, D. W.; Roca, J. DNA knots reveal a chiral organization of DNA in phage capsids. Proceedings of the National Academy of Sciences 2005, 102, 9165--9169, DOI: doi:10.1073/pnas.0409323102

    2005 doi

  6. [14]

    DNA knots occur in intracellular chromatin

    Valdés, A.; Segura, J.; Dyson, S.; Martínez-García, B.; Roca, J. DNA knots occur in intracellular chromatin. Nucleic Acids Research 2018, 46, 650--660, DOI: doi:10.1093/nar/gkx1137

    2018 doi

  7. [15]

    F.; Perkocha, L.; Calendar, R.; Wang, J

    Liu, L. F.; Perkocha, L.; Calendar, R.; Wang, J. C. Knotted DNA from bacteriophage capsids. Proceedings of the National Academy of Sciences 1981, 78, 5498--5502, DOI: doi:10.1073/pnas.78.9.5498

    1981 doi

  8. [16]

    Spontaneous Knotting and Unknotting of Flexible Linear Polymers : Equilibrium and Kinetic Aspects

    Tubiana, L.; Rosa, A.; Fragiacomo, F.; Micheletti, C. Spontaneous Knotting and Unknotting of Flexible Linear Polymers : Equilibrium and Kinetic Aspects . Macromolecules 2013, 46, 3669--3678, DOI: doi:10.1021/ma4002963

    2013 doi

  9. [17]

    W.; Witteveen, M

    Plesa, C.; Verschueren, D.; Pud, S.; van der Torre, J.; Ruitenberg, J. W.; Witteveen, M. J.; Jonsson, M. P.; Grosberg, A. Y.; Rabin, Y.; Dekker, C. Direct observation of DNA knots using a solid-state nanopore. Nature Nanotechnology 2016, 11, 1093--1097, DOI: doi:10.1038/nnano.2016.153

    2016 doi

  10. [18]

    Pore translocation of knotted DNA rings

    Suma, A.; Micheletti, C. Pore translocation of knotted DNA rings. Proceedings of the National Academy of Sciences 2017, 114, E2991--E2997, DOI: doi:10.1073/pnas.1701321114

    2017 doi

  11. [19]

    R.; Narsimhan, V.; Soh, B

    Klotz, A. R.; Narsimhan, V.; Soh, B. W.; Doyle, P. S. Dynamics of DNA Knots during Chain Relaxation . Macromolecules 2017, 50, 4074--4082, DOI: doi:10.1021/acs.macromol.7b00287

    2017 doi

  12. [20]

    Ma, Z.; Dorfman, K. D. Diffusion of Knotted DNA Molecules in Nanochannels in the Extended de Gennes Regime . Macromolecules 2021, 54, 4211--4218, DOI: doi:10.1021/acs.macromol.1c00143

    2021 doi

  13. [21]

    J.; Kern, J

    Rawdon, E. J.; Kern, J. C.; Piatek, M.; Plunkett, P.; Stasiak, A.; Millett, K. C. Effect of Knotting on the Shape of Polymers . Macromolecules 2008, 41, 8281--8287, DOI: doi:10.1021/ma801389c

    2008 doi

  14. [22]

    C.; Plunkett, P.; Piatek, M.; Rawdon, E

    Millett, K. C.; Plunkett, P.; Piatek, M.; Rawdon, E. J.; Stasiak, A. Effect of knotting on polymer shapes and their enveloping ellipsoids. The Journal of Chemical Physics 2009, 130, 165104, DOI: doi:10.1063/1.3117923

    2009 doi

  15. [23]

    Effect of Topology on the Conformations of Ring Polymers

    Lang, M.; Fischer, J.; Sommer, J.-U. Effect of Topology on the Conformations of Ring Polymers . Macromolecules 2012, 45, 7642--7648, DOI: doi:10.1021/ma300942a

    2012 doi

  16. [24]

    Baiesi, M.; Orlandini, E.; Stella, A. L. Knotted Globular Ring Polymers : How Topology Affects Statistics and Thermodynamics . Macromolecules 2014, 47, 8466--8476, DOI: doi:10.1021/ma5020287

    2014 doi

  17. [25]

    Topological origin of reptation: a collective motion of local knots

    Iwata, K. Topological origin of reptation: a collective motion of local knots. Macromolecules 1991, 24, 1107--1116

    1991

  18. [26]

    How Local Flexibility Affects Knot Positioning in Ring Polymers

    Orlandini, E.; Baiesi, M.; Zonta, F. How Local Flexibility Affects Knot Positioning in Ring Polymers . Macromolecules 2016, 49, 4656--4662, DOI: doi:10.1021/acs.macromol.6b00712

    2016 doi

  19. [27]

    B.; Marenda, M.; Micheletti, C.; Likos, C

    Weiss, L. B.; Marenda, M.; Micheletti, C.; Likos, C. N. Hydrodynamics and Filtering of Knotted Ring Polymers in Nanochannels . Macromolecules 2019, 52, 4111--4119, DOI: doi:10.1021/acs.macromol.9b00516

    2019 doi

  20. [28]

    Tuning Knotted Copolyelectrolyte Conformations via Solution Properties

    Tagliabue, A.; Micheletti, C.; Mella, M. Tuning Knotted Copolyelectrolyte Conformations via Solution Properties . Macromolecules 2022, 55, 10761--10772, DOI: doi:10.1021/acs.macromol.2c01933

    2022 doi

  21. [29]

    Dai, L.; Doyle, P. S. Effects of Intrachain Interactions on the Knot Size of a Polymer . Macromolecules 2016, 49, 7581--7587, DOI: doi:10.1021/acs.macromol.6b01653

    2016 doi

  22. [30]

    Controlling Knot Interactions through Confinement

    Rusková, R.; Tubiana, L.; Potestio, R.; Račko, D. Controlling Knot Interactions through Confinement . Macromolecules 2025, 58, 6718--6731, DOI: doi:10.1021/acs.macromol.4c02977

    2025 doi

  23. [31]

    F.; Baskaran, A

    Hagan, M. F.; Baskaran, A. Emergent self-organization in active materials. Current Opinion in Cell Biology 2016, 38, 74--80, DOI: doi:https://doi.org/10.1016/j.ceb.2016.02.020, Cell architecture

    2016 doi

  24. [32]

    Active matter at the interface between materials science and cell biology

    Needleman, D.; Dogic, Z. Active matter at the interface between materials science and cell biology. Nature Reviews Materials 2017, 2, 1--14, DOI: doi:10.1038/natrevmats.2017.48

    2017 doi

  25. [33]

    Collective motion

    Vicsek, T.; Zafeiris, A. Collective motion. Physics reports 2012, 517, 71--140

    2012

  26. [34]

    Bird Flocks as Condensed Matter

    Cavagna, A.; Giardina, I. Bird Flocks as Condensed Matter . Annual Review of Condensed Matter Physics 2014, 5, 183--207, DOI: doi:10.1146/annurev-conmatphys-031113-133834

    2014 doi

  27. [35]

    C.; Joanny, J.-F.; Ramaswamy, S.; Liverpool, T

    Marchetti, M. C.; Joanny, J.-F.; Ramaswamy, S.; Liverpool, T. B.; Prost, J.; Rao, M.; Simha, R. A. Hydrodynamics of soft active matter. Reviews of modern physics 2013, 85, 1143--1189

    2013

  28. [36]

    Dynamical Clustering and Phase Separation in Suspensions of Self - Propelled Colloidal Particles

    Buttinoni, I.; Bialké, J.; Kümmel, F.; Löwen, H.; Bechinger, C.; Speck, T. Dynamical Clustering and Phase Separation in Suspensions of Self - Propelled Colloidal Particles . Physical Review Letters 2013, 110, 238301, DOI: doi:10.1103/PhysRevLett.110.238301

    2013 doi

  29. [37]

    E.; Tailleur, J

    Cates, M. E.; Tailleur, J. Motility- Induced Phase Separation . Annual Review of Condensed Matter Physics 2015, 6, 219--244, DOI: doi:10.1146/annurev-conmatphys-031214-014710

    2015 doi

  30. [38]

    G.; Gompper, G

    Winkler, R. G.; Gompper, G. The physics of active polymers and filaments. The Journal of Chemical Physics 2020, 153, 040901, DOI: doi:10.1063/5.0011466

    2020 doi

  31. [39]

    Cilia- Like Beating of Active Microtubule Bundles

    Sanchez, T.; Welch, D.; Nicastro, D.; Dogic, Z. Cilia- Like Beating of Active Microtubule Bundles . Science 2011, 333, 456--459, DOI: doi:10.1126/science.1203963

    2011 doi

  32. [40]

    H.; Shitaka, Y.; Tanaka, D.; Yoshikawa, K.; Chaté, H.; Oiwa, K

    Sumino, Y.; Nagai, K. H.; Shitaka, Y.; Tanaka, D.; Yoshikawa, K.; Chaté, H.; Oiwa, K. Large-scale vortex lattice emerging from collectively moving microtubules. Nature 2012, 483, 448--452, DOI: doi:10.1038/nature10874

    2012 doi

  33. [41]

    I.; Bhamla, M

    Ozkan-Aydin, Y.; Goldman, D. I.; Bhamla, M. S. Collective dynamics in entangled worm and robot blobs. Proceedings of the National Academy of Sciences 2021, 118, e2010542118

    2021

  34. [42]

    P.; Locatelli, E.; Valeriani, C

    Miranda, J. P.; Locatelli, E.; Valeriani, C. Self-organized states from solutions of active ring polymers in bulk and under confinement. Journal of Chemical Theory and Computation 2023, 20, 1636--1645

    2023

  35. [43]

    P.; Tuazon, H.; Kaufman, E.; Chakrabortty, T.; Qin, D.; Dunkel, J.; Bhamla, M

    Patil, V. P.; Tuazon, H.; Kaufman, E.; Chakrabortty, T.; Qin, D.; Dunkel, J.; Bhamla, M. S. Ultrafast reversible self-assembly of living tangled matter. Science 2023, 380, 392--398

    2023

  36. [44]

    K.; Cammann, J.; Mazza, M

    Faluweki, M. K.; Cammann, J.; Mazza, M. G.; Goehring, L. Active spaghetti: collective organization in cyanobacteria. Physical review letters 2023, 131, 158303

    2023

  37. [45]

    Phase Separation by Entanglement of Active Polymerlike Worms

    Deblais, A.; Maggs, A.; Bonn, D.; Woutersen, S. Phase Separation by Entanglement of Active Polymerlike Worms . Physical Review Letters 2020, 124, 208006, DOI: doi:10.1103/PhysRevLett.124.208006

    2020 doi

  38. [46]

    Schaller, V.; Weber, C.; Semmrich, C.; Frey, E.; Bausch, A. R. Polar patterns of driven filaments. Nature 2010, 467, 73--77

    2010

  39. [47]

    Globulelike Conformation and Enhanced Diffusion of Active Polymers

    Bianco, V.; Locatelli, E.; Malgaretti, P. Globulelike Conformation and Enhanced Diffusion of Active Polymers . Physical Review Letters 2018, 121, 217802, DOI: doi:10.1103/PhysRevLett.121.217802

    2018 doi

  40. [48]

    R.; Ramírez, J

    Tejedor, A. R.; Ramírez, J. Reptation of Active Entangled Polymers . Macromolecules 2019, 52, 8788--8792, DOI: doi:10.1021/acs.macromol.9b01994

    2019 doi

  41. [49]

    Rheology of Entangled Active Polymer - Like T

    Deblais, A.; Woutersen, S.; Bonn, D. Rheology of Entangled Active Polymer - Like T . Tubifex Worms . Physical Review Letters 2020, 124, 188002, DOI: doi:10.1103/PhysRevLett.124.188002

    2020 doi

  42. [50]

    Giant activity-induced elasticity in entangled polymer solutions

    Breoni, D.; Kurzthaler, C.; Liebchen, B.; Löwen, H.; Mandal, S. Giant activity-induced elasticity in entangled polymer solutions. Nature Communications 2025, 16, 5305, DOI: doi:10.1038/s41467-025-60210-9

    2025 doi

  43. [51]

    H.; Dogic, Z.; Nakamura, F.; Bendix, P

    Koenderink, G. H.; Dogic, Z.; Nakamura, F.; Bendix, P. M.; MacKintosh, F. C.; Hartwig, J. H.; Stossel, T. P.; Weitz, D. A. An active biopolymer network controlled by molecular motors. Proceedings of the National Academy of Sciences 2009, 106, 15192--15197, DOI: doi:10.1073/pna...

    2009 doi

  44. [52]

    Bertrand, O. J. N.; Fygenson, D. K.; Saleh, O. A. Active, motor-driven mechanics in a DNA gel. Proceedings of the National Academy of Sciences 2012, 109, 17342--17347, DOI: doi:10.1073/pnas.1208732109

    2012 doi

  45. [53]

    T.; Giuseppone, N

    Li, Q.; Fuks, G.; Moulin, E.; Maaloum, M.; Rawiso, M.; Kulic, I.; Foy, J. T.; Giuseppone, N. Macroscopic contraction of a gel induced by the integrated motion of light-driven molecular motors. Nature Nanotechnology 2015, 10, 161--165, DOI: doi:10.1038/nnano.2014.315

    2015 doi

  46. [54]

    E.; Aufderhorst-Roberts, A.; Koenderink, G

    Burla, F.; Mulla, Y.; Vos, B. E.; Aufderhorst-Roberts, A.; Koenderink, G. H. From mechanical resilience to active material properties in biopolymer networks. Nature Reviews Physics 2019, 1, 249--263, DOI: doi:10.1038/s42254-019-0036-4

    2019 doi

  47. [55]

    A.; Fedosov, D

    Sciortino, A.; Faizi, H. A.; Fedosov, D. A.; Frechette, L.; Vlahovska, P. M.; Gompper, G.; Bausch, A. R. Active membrane deformations of a minimal synthetic cell. Nature Physics 2025, 1--9

    2025

  48. [56]

    P.; Bhamla, M

    Deblais, A.; Prathyusha, K.; Sinaasappel, R.; Tuazon, H.; Tiwari, I.; Patil, V. P.; Bhamla, M. S. Worm blobs as entangled living polymers: from topological active matter to flexible soft robot collectives. Soft Matter 2023, 19, 7057--7069

    2023

  49. [57]

    D.; Lewis, J.; Raff, M.; Roberts, K.; Walter, P

    Alberts, B.; Bray, D.; Hopkin, K.; Johnson, A. D.; Lewis, J.; Raff, M.; Roberts, K.; Walter, P. Essential Cell Biology ; Garland Science, 2015

    2015

  50. [58]

    N.; Kremer, K

    Smrek, J.; Chubak, I.; Likos, C. N.; Kremer, K. Active topological glass. Nature communications 2020, 11, 26

    2020

  51. [59]

    Topology- Based Detection and Tracking of Deadlocks Reveal Aging of Active Ring Melts

    Micheletti, C.; Chubak, I.; Orlandini, E.; Smrek, J. Topology- Based Detection and Tracking of Deadlocks Reveal Aging of Active Ring Melts . ACS Macro Letters 2024, 13, 124--129, DOI: doi:10.1021/acsmacrolett.3c00567

    2024 doi

  52. [60]

    Activity-driven polymer knotting for macromolecular topology engineering

    Li, J.-X.; Wu, S.; Hao, L.-L.; Lei, Q.-L.; Ma, Y.-Q. Activity-driven polymer knotting for macromolecular topology engineering. Science Advances 2024, 10, eadr0716, DOI: doi:10.1126/sciadv.adr0716

    2024 doi

  53. [61]

    Upsurge of Spontaneous Knotting in Polar Diblock Active Polymers

    Vatin, M.; Orlandini, E.; Locatelli, E. Upsurge of Spontaneous Knotting in Polar Diblock Active Polymers . Physical Review Letters 2025, 134, 168301, DOI: doi:10.1103/PhysRevLett.134.168301

    2025 doi

  54. [62]

    Knotting and adsorption of end-grafted active polymers

    Shen, Y.-F.; Luo, M.-B. Knotting and adsorption of end-grafted active polymers. Soft Matter 2025, 21, 1873--1883, DOI: doi:10.1039/D4SM01355G

    2025 doi

  55. [63]

    Activity- Induced Collapse and Arrest of Active Polymer Rings

    Locatelli, E.; Bianco, V.; Malgaretti, P. Activity- Induced Collapse and Arrest of Active Polymer Rings . Physical Review Letters 2021, 126, 097801, DOI: doi:10.1103/PhysRevLett.126.097801

    2021 doi

  56. [64]

    Y.; Feigel, A.; Rabin, Y

    Grosberg, A. Y.; Feigel, A.; Rabin, Y. Flory-type theory of a knotted ring polymer. Physical Review E 1996, 54, 6618--6622, DOI: doi:10.1103/PhysRevE.54.6618

    1996 doi

  57. [65]

    Ideal Knots ; World Scientific, 1998

    Stasiak, A.; Katritch, V. Ideal Knots ; World Scientific, 1998

    1998

  58. [66]

    S.; Kremer, K

    Grest, G. S.; Kremer, K. Molecular dynamics simulation for polymers in the presence of a heat bath. Physical Review A 1986, 33, 3628--3631, DOI: doi:10.1103/PhysRevA.33.3628

    1986 doi

  59. [67]

    P.; Aktulga, H

    Thompson, A. P.; Aktulga, H. M.; Berger, R.; Bolintineanu, D. S.; Brown, W. M.; Crozier, P. S.; in 't Veld, P. J.; Kohlmeyer, A.; Moore, S. G.; Nguyen, T. D.; Shan, R.; Stevens, M. J.; Tranchida, J.; Trott, C.; Plimpton, S. J. LAMMPS - a flexible simulation tool for particle-b...

    2022

  60. [68]

    E.; Elgeti, J.; Gompper, G

    Isele-Holder, R. E.; Elgeti, J.; Gompper, G. Self-propelled worm-like filaments: spontaneous spiral formation, structure, and dynamics. Soft matter 2015, 11, 7181--7190

    2015

  61. [69]

    K.; Singh, S

    Anand, S. K.; Singh, S. P. Structure and dynamics of a self-propelled semiflexible filament. Physical Review E 2018, 98, 042501

    2018

  62. [70]

    P.; Martín-Roca, J.; Malgaretti, P.; Locatelli, E.; Valeriani, C.; Fernandez, D

    Janzen, G.; Miranda, J. P.; Martín-Roca, J.; Malgaretti, P.; Locatelli, E.; Valeriani, C.; Fernandez, D. A. M. Active polymer behavior in two dimensions: A comparative analysis of tangential and push–pull models. The Journal of Chemical Physics 2025, 162, 114905

    2025

  63. [71]

    Oberti, C.; Ricca, R. L. On torus knots and unknots. Journal of Knot Theory and Its Ramifications 2016, DOI: doi:10.1142/S021821651650036X

    2016 doi

  64. [72]

    W.; Bohr, J

    Olsen, K. W.; Bohr, J. A principle for ideal torus knots. Europhysics Letters 2013, 103, 30002, DOI: doi:10.1209/0295-5075/103/30002

    2013 doi

  65. [73]

    G.; Dubochet, J.; Stasiak, A

    Katritch, V.; Bednar, J.; Michoud, D.; Scharein, R. G.; Dubochet, J.; Stasiak, A. Geometry and physics of knots. Nature 1996, 384, 142--145, DOI: doi:10.1038/384142a0

    1996 doi

  66. [74]

    De Gennes, P. G. Tight knots. Macromolecules 1984, 17, 703--704

    1984

  67. [75]

    Y.; Rabin, Y

    Grosberg, A. Y.; Rabin, Y. Metastable Tight Knots in a Wormlike Polymer . Physical Review Letters 2007, 99, 217801, DOI: doi:10.1103/PhysRevLett.99.217801

    2007 doi

  68. [76]

    B.; Doyle, P

    Dai, L.; Renner, C. B.; Doyle, P. S. Metastable Tight Knots in Semiflexible Chains . Macromolecules 2014, 47, 6135--6140, DOI: doi:10.1021/ma501585x

    2014 doi

  69. [77]

    Liebetreu, M.; Likos, C. N. Hydrodynamic inflation of ring polymers under shear. Communications Materials 2020, 1, 4, DOI: doi:10.1038/s43246-019-0006-5

    2020 doi

  70. [78]

    Heteropolymer Design and Folding of Arbitrary Topologies Reveals an Unexpected Role of Alphabet Size on the Knot Population

    Cardelli, C.; Tubiana, L.; Bianco, V.; Nerattini, F.; Dellago, C.; Coluzza, I. Heteropolymer Design and Folding of Arbitrary Topologies Reveals an Unexpected Role of Alphabet Size on the Knot Population . Macromolecules 2018, 51, 8346--8356, DOI: doi:10.1021/acs.macromol.8b01359

    2018 doi

  71. [79]

    Local Polar and Long - Range Isotropic Activity Assisted Swelling and Collapse Dynamics of an Active Ring Polymer

    Kumar, S.; Thakur, S. Local Polar and Long - Range Isotropic Activity Assisted Swelling and Collapse Dynamics of an Active Ring Polymer . Macromolecules 2023, 56, 5229--5236, DOI: doi:10.1021/acs.macromol.3c00384

    2023 doi

  72. [80]

    How a protein searches for its site on DNA : the mechanism of facilitated diffusion

    Mirny, L.; Slutsky, M.; Wunderlich, Z.; Tafvizi, A.; Leith, J.; Kosmrlj, A. How a protein searches for its site on DNA : the mechanism of facilitated diffusion. Journal of Physics A: Mathematical and Theoretical 2009, 42, 434013, DOI: doi:10.1088/1751-8113/42/43/434013

    2009 doi

  73. [81]

    SCOPE—Selective Capture Of Metals by Polymeric spongEs

    Suter, D. M. Transcription Factors and DNA Play Hide and Seek . Trends in Cell Biology 2020, 30, 491--500, DOI: doi:10.1016/j.tcb.2020.03.003 mcitethebibliography acknots.tex0000664000000000000000000015357215042710521011754 0ustar rootroot [journal=mamobx,manuscript=article] a...

    2020 doi

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.