pith. sign in

arxiv: 2601.09908 · v3 · pith:7C6YODLRnew · submitted 2026-01-14 · ❄️ cond-mat.soft

Collapse of a single polymer chain: Effects of chain stiffness and attraction range

Yanyan Zhu , Haim Diamant , David Andelman This is my paper

Pith reviewed 2026-05-21 15:30 UTC · model grok-4.3

classification ❄️ cond-mat.soft
keywords polymer collapsechain stiffnesspersistence lengthattraction rangeMonte Carlotheta temperaturesingle chain contraction
0
0 comments X

The pith

The competition between persistence length and attraction range decides whether a polymer collapses sharply or gradually with temperature.

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

This paper explores the collapse of a single polymer chain by examining how its stiffness interacts with the range of attractions between its monomers. Through simulations, it finds that when the persistence length is greater than the attraction range, the chain collapses in a sharp manner as temperature decreases. If the persistence length is smaller than the attraction range, the chain instead contracts in a gradual fashion. This gradual contraction can continue even as the chain gets longer and may hold for infinitely long chains. The work also reveals that increasing stiffness raises the transition temperature for short attraction ranges but lowers it for long ranges.

Core claim

The paper demonstrates that the competition between the persistence length, l_p, and the range of attraction, r_c, determines whether the chain's collapse behavior resembles that of flexible chains or stiff ones. When l_p is larger than r_c, the chain collapses sharply with decreasing temperature, whereas if l_p is smaller than r_c, it contracts gradually. Notably, in the regime of small l_p and large r_c, this rounding into a gradual compaction persists upon increasing the chain length and may remain in place in the limit of infinite chain length. Furthermore, for small r_c, the transition temperature increases with l_p, whereas for large r_c the theta-temperature decreases with l_p.

What carries the argument

The key mechanism is the direct comparison of persistence length l_p to attraction range r_c, studied via Monte Carlo simulations, to classify the collapse as sharp or gradual.

If this is right

  • When l_p exceeds r_c the collapse becomes sharp like in stiff chains.
  • When l_p is less than r_c the contraction is gradual and may not sharpen with longer chains.
  • Stiffness promotes collapse for small r_c but suppresses it for large r_c.
  • The theta-temperature increases with l_p for small r_c and decreases for large r_c.

Where Pith is reading between the lines

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

  • This distinction could account for observed differences in contraction between single-stranded RNA and double-stranded DNA.
  • Similar parameter competition might influence collapse transitions in other chain-like systems such as proteins.
  • Adjusting these lengths could allow control over whether a polymer undergoes a first-order like transition or a smooth crossover.

Load-bearing premise

The simulations using the pruned-enriched Rosenbluth method correctly sample the equilibrium properties and finite size effects for the studied lengths, supporting extrapolation to infinite chains.

What would settle it

Observing the collapse behavior in much longer chains where persistence length is smaller than attraction range to check if the transition stays gradual or eventually sharpens.

Figures

Figures reproduced from arXiv: 2601.09908 by David Andelman, Haim Diamant, Yanyan Zhu.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: , for lp = 0 and rc = 3, chains of length N = 1,500 (1/N = 6.7 × 10−4 ) are already collapsed at T ∗ = 60, whereas the shorter chains are not. This observation sug￾gests that the collapse transition temperature depends on the chain length. As the chain length increases, the tran￾sition temperature is expected to approach the θ-point for flexible chains in the thermodynamic limit (N → ∞). The chain length a… view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
read the original abstract

Chain-like macromolecules in solution, whether biological or synthetic, transform from an extended conformation to a compact one when temperature or other system parameters change. This collapse transition is relevant in various phenomena, including DNA condensation, protein folding, and the behavior of polymers in solution. We investigate the interplay of chain stiffness and range of attraction between monomers in the collapse of a single polymer chain. We use Monte Carlo simulations based on the pruned-enriched Rosenbluth method. We demonstrate that the competition between the persistence length, l_p, and the range of attraction, r_c, determines whether the chain's collapse behavior resembles that of flexible chains or stiff ones. When l_p is larger than r_c, the chain collapses sharply with decreasing temperature, whereas if l_p is smaller than r_c, it contracts gradually. Notably, in the regime of small l_p and large r_c, this rounding into a gradual compaction persists upon increasing the chain length and may remain in place in the limit of infinite chain length. Furthermore, for small r_c, the transition temperature (theta-temperature) increases with l_p, whereas for large r_c the theta-temperature decreases with l_p. Thus, stiffness promotes collapse for small r_c but suppresses it for large r_c. Our findings are in agreement with recent experiments on the contraction of single-stranded RNA as compared to double-stranded DNA, and provide valuable insights for understanding polymer collapse and the essential polymer parameters affecting it.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript uses pruned-enriched Rosenbluth Monte Carlo simulations to examine the collapse of a single polymer chain as a function of persistence length lp and attraction range rc. It reports that lp > rc produces a sharp collapse upon cooling while lp < rc yields gradual compaction; the latter rounding is asserted to survive increasing chain length N and possibly to persist at N→∞. The theta temperature is found to rise with lp for small rc but to fall with lp for large rc. Results are compared to single-molecule experiments on ssRNA versus dsDNA.

Significance. If the central distinction between regimes and the infinite-N claim hold, the work supplies a concrete parameter-based criterion (lp versus rc) that rationalizes why some polymers collapse sharply and others gradually. The direct-simulation approach with no adjustable parameters after the model definition is a methodological strength and the experimental contact adds relevance for biological macromolecules.

major comments (1)
  1. [Abstract and results] Abstract and main results: the assertion that gradual compaction in the lp < rc regime 'persists upon increasing the chain length and may remain in place in the limit of infinite chain length' rests on finite-N data. No finite-size scaling collapse, Binder-cumulant analysis, or explicit demonstration that the effective transition width saturates (rather than narrowing as N^{-1/2} or N^{-1}) is described. Because this extrapolation underpins the claimed qualitative distinction between the two regimes, it constitutes a load-bearing gap.
minor comments (2)
  1. [Model] Notation for the attraction range rc and the precise functional form of the potential should be stated explicitly in the model section to allow direct reproduction.
  2. [Methods] The range of chain lengths N actually simulated and the number of independent runs per (lp, rc, T) point should be tabulated or stated clearly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We appreciate the positive assessment of the methodological approach and the experimental relevance. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract and results] Abstract and main results: the assertion that gradual compaction in the lp < rc regime 'persists upon increasing the chain length and may remain in place in the limit of infinite chain length' rests on finite-N data. No finite-size scaling collapse, Binder-cumulant analysis, or explicit demonstration that the effective transition width saturates (rather than narrowing as N^{-1/2} or N^{-1}) is described. Because this extrapolation underpins the claimed qualitative distinction between the two regimes, it constitutes a load-bearing gap.

    Authors: We agree that the claim concerning the persistence of gradual compaction at large N (and possibly N→∞) in the lp < rc regime rests on finite-N data and that the manuscript does not include a formal finite-size scaling analysis, Binder cumulant crossings, or an explicit demonstration that the effective transition width saturates. Our simulations for increasing N showed that the compaction remains gradual without sharpening in this regime, in contrast to the lp > rc case, but we acknowledge that this does not constitute rigorous proof of the thermodynamic-limit behavior. In the revised manuscript we will add a dedicated discussion of finite-size effects, including the scaling of the specific-heat peak width with N and, where feasible, Binder cumulant data to substantiate that the rounding persists rather than narrowing as N^{-1/2} or N^{-1}. revision: yes

Circularity Check

0 steps flagged

No circularity: results from direct Monte Carlo sampling

full rationale

The paper reports equilibrium statistics obtained via pruned-enriched Rosenbluth Monte Carlo sampling for finite chain lengths N. All reported behaviors (sharp vs. gradual compaction, dependence of theta temperature on lp and rc) are direct numerical outputs rather than quantities derived from equations that reduce to the simulation inputs by construction. No self-citations, fitted parameters renamed as predictions, or ansatzes are invoked to support the central claims; the infinite-N extrapolation is presented as a possible inference from the observed trend, not as a mathematical identity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the accuracy of the pruned-enriched Rosenbluth Monte Carlo sampling for the chosen polymer model with variable stiffness and finite-range attractions; limited information is available from the abstract alone.

axioms (1)
  • domain assumption The pruned-enriched Rosenbluth method provides unbiased sampling of the equilibrium ensemble for the polymer model.
    Invoked by basing all results on Monte Carlo simulations using this method.

pith-pipeline@v0.9.0 · 5790 in / 1323 out tokens · 87021 ms · 2026-05-21T15:30:02.879432+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages

  1. [1]

    Y.; Khokhlov, A

    Grosberg, A. Y.; Khokhlov, A. R.Statistical Physics of Macromolecules; AIP Press, New York,1994

    work page 1994

  2. [2]

    Soret Coefficient of Poly(N- isopropylacrylamide)/Water in the Vicinity of Coil- Globule Transition Temperature.Macromolecules2005, 38, 4554-4556

    Kita, R.; Wiegand, S. Soret Coefficient of Poly(N- isopropylacrylamide)/Water in the Vicinity of Coil- Globule Transition Temperature.Macromolecules2005, 38, 4554-4556

    work page

  3. [3]

    R.; Hubbuch, J

    Frerix, A.; Schonewald, M.; Geilenkirchen, P.; M¨ uller, M.; Kula, M. R.; Hubbuch, J. Exploitation of the Coil-Globule Plasmid DNA Transition Induced by Small Changes in Temperature, pH, Salt, and Poly(ethylene glycol) Compositions for Directed Partitioning in Aque- ous Two-Phase Systems.Langmuir2006,22, 4282-4290

    work page

  4. [4]

    Tanaka, F.; Koga, T.; Kojima, H.; Winnik, F. M. Temperature- and Tension-Induced Coil-Globule Tran- sition of Poly(N-isopropylacrylamide) Chains in Water and Mixed Solvent of Water/Methanol.Macromolecules 2009,42, 1321-1330

    work page 2009

  5. [5]

    Coil-Globule Transition of Poly(methyl methacrylate) in a Mixed Solvent.Phys

    Nakata, M. Coil-Globule Transition of Poly(methyl methacrylate) in a Mixed Solvent.Phys. Rev. E1995, 51, 5770

    work page

  6. [6]

    Coil-Globule Transition in the Denatured State of a Small Protein.Proc

    Sherman, E.; Haran, G. Coil-Globule Transition in the Denatured State of a Small Protein.Proc. Natl. Acad. Sci. U.S.A.2006,103, 11539-11543

    work page 2006

  7. [7]

    First Observation of Two-Stage Collapsing Kinetics of a Single Synthetic Polymer Chain.Phys

    Xu, J.; Zhu, Z.; Luo, S.; Wu, C.; Liu, S. First Observation of Two-Stage Collapsing Kinetics of a Single Synthetic Polymer Chain.Phys. Rev. Lett.2006,96, 027802

    work page 2006

  8. [8]

    Stockmayer, W. H. Problems of the Statistical Thermo- dynamics of Dilute Polymer Solutions.Makromol. Chem. 1960,35, 54-74

    work page 1960

  9. [9]

    T.; Swislow, G.; Tanaka, T

    Nishio, I.; Sun, S. T.; Swislow, G.; Tanaka, T. First Ob- servation of the Coil-Globule Transition in a Single Poly- mer Chain.Nature1979,281, 208-209

    work page

  10. [10]

    T.; Nishio, I.; Tanaka, T

    Swislow, G.; Sun, S. T.; Nishio, I.; Tanaka, T. Coil- Globule Phase Transition in a Single Polystyrene Chain in Cyclohexane.Phys. Rev. Lett.1980,44, 796

    work page 1980

  11. [11]

    Lifshitz, I. M. Some Problems of the Statistical Theory of Biopolymers.Zh. Eksp. Teor. Fiz.1968,55, 2408-2422. (Engl. Transl.: Sov. Phys. JETP1969,28, 1280-1286)

    work page 1968

  12. [12]

    M.; Grosberg, A

    Lifshitz, I. M.; Grosberg, A. Y.; Khokhlov, A. R. Some Problems of the Statistical Physics of Polymer Chains with Volume Interaction.Rev. Mod. Phys.1978,50, 683- 713

    work page 1978

  13. [13]

    Y.; Kuznetsov, D

    Grosberg, A. Y.; Kuznetsov, D. V. Quantitative Theory of the Globule-to-Coil Transition. 1. Link Density Distri- bution in a Globule and Its Radius of Gyration.Macro- molecules1992,25, 1970-1979

    work page 1970

  14. [14]

    Y.; Kuznetsov, D

    Grosberg, A. Y.; Kuznetsov, D. V. Quantitative The- ory of the Globule-to-Coil Transition. 4. Comparison of Theoretical Results with Experimental Data.Macro- molecules1992,25, 1996-2003

    work page 1996

  15. [15]

    Globule-to-Coil Transition of a Single Homopolymer Chain in Solution.Phys

    Wu, C.; Wang, X. Globule-to-Coil Transition of a Single Homopolymer Chain in Solution.Phys. Rev. Lett.1998, 80, 4092-4094

    work page 1998

  16. [16]

    First Observation of the Molten Globule State of a Single Homopolymer Chain.Phys

    Wu, C.; Zhou, S. First Observation of the Molten Globule State of a Single Homopolymer Chain.Phys. Rev. Lett. 1996,77, 3053

    work page 1996

  17. [17]

    M.; Kayaman, N

    Baysal, B. M.; Kayaman, N. Coil-Globule Transition of Poly(methyl methacrylate) by Intrinsic Viscosity.J. Chem. Phys.1998,109, 8701

    work page 1998

  18. [18]

    How Many Stages Are in the Coil-to-Globule Transition of Linear Homopolymer Chains in a Dilute Solution?Macromolecules2007,40, 4750

    Ye, X.; Lu, Y.; Shen, L.; Ding, Y.; Liu, S.; Zhang, G.; Wu, C. How Many Stages Are in the Coil-to-Globule Transition of Linear Homopolymer Chains in a Dilute Solution?Macromolecules2007,40, 4750

    work page

  19. [19]

    Monitoring Coil-Globule Transitions of Thermore- sponsive Polymers by Using NMR Solvent Relaxation

    Chakraborty, I.; Mukherjee, K.; De, P.; Bhattacharyya, R. Monitoring Coil-Globule Transitions of Thermore- sponsive Polymers by Using NMR Solvent Relaxation. J. Phys. Chem. B2018,122, 6094

    work page

  20. [20]

    F.; Siggia, E

    Bustamante, C.; Marko, J. F.; Siggia, E. D.; Smith, S. Entropic Elasticity ofλ-Phage DNA.Science1994,265, 1599-1600

    work page

  21. [21]

    Gerrits, L.; Hammink, R.; Kouwer, P. H. J. Semiflexible Polymer Scaffolds: An Overview of Conjugation Strate- gies.Polym. Chem.2021,12, 1362-1392

    work page 2021

  22. [22]

    R.; Giacometti, A

    Skrbic, T.; Banavar, J. R.; Giacometti, A. Chain Stiffness Bridges Conventional Polymer and Biomolecular Phases. J. Chem. Phys.2019,151, 174901

    work page 2019

  23. [23]

    R.; Piazza, R.; Maritan, A.; Giacometti, A

    Arcangeli, T.; Skrbic, T.; Azote, S.; Marcato, D.; Rosa, A.; Banavar, J. R.; Piazza, R.; Maritan, A.; Giacometti, A. Phase Behavior and Self-Assembly of Semiflexible Polymers in Poor-Solvent Solutions.Macro- molecules2024,57, 8940-8955

    work page

  24. [24]

    Structure of Toroidal DNA Collapsed Inside the Phage Capsid.Proc

    Leforestier, A.; Livolant, F. Structure of Toroidal DNA Collapsed Inside the Phage Capsid.Proc. Natl. Acad. Sci. U.S.A.2009,106, 9157-9162

    work page 2009

  25. [25]

    Protein-DNA Interactions Determine the Shapes of DNA Toroids Condensed in Virus Capsids.Biophys

    Leforestier, A.; Siber, A.; Livolant, F.; Podgornik, R. Protein-DNA Interactions Determine the Shapes of DNA Toroids Condensed in Virus Capsids.Biophys. J.2011, 100, 2209-2216

    work page 2011

  26. [26]

    T.; Schnabel, S.; Landau, D

    Seaton, D. T.; Schnabel, S.; Landau, D. P.; Bachmann, M. From Flexible to Stiff: Systematic Analysis of Struc- tural Phases for Single Semiflexible Polymers.Phys. Rev. Lett.2013,110, 028103

    work page 2013

  27. [27]

    Knots as a Topological Order Pa- rameter for Semiflexible Polymers.Phys

    Marenz, M.; Janke, W. Knots as a Topological Order Pa- rameter for Semiflexible Polymers.Phys. Rev. Lett.2016, 116, 128301

    work page 2016

  28. [28]

    Knots Are Generic Stable Phases in Semiflexible Polymers

    Majumder, S.; Marenz, M.; Paul, S.; Janke, W. Knots Are Generic Stable Phases in Semiflexible Polymers. Macromolecules2021,54, 5321-5334

    work page

  29. [29]

    Unified View on the Mean-Field Or- der of Coil-Globule Transition.ACS Macro Lett.2013, 2, 952-954

    Yang, D.; Wang, Q. Unified View on the Mean-Field Or- der of Coil-Globule Transition.ACS Macro Lett.2013, 2, 952-954

    work page 2013

  30. [30]

    Wang, Z. G. 50th Anniversary Perspective: Polymer Conformation – A Pedagogical Review.Macromolecules 2017,50, 9073-9114

    work page 2017

  31. [31]

    B.; Zimm, B

    Post, C. B.; Zimm, B. H. Internal Condensation of a Single DNA Molecule.Biopolymers1979,18, 1487-1501

    work page

  32. [32]

    The Single Polymer Chain Phase Diagram: New Insights from a New Sim- ulation Method.J

    Rampf, F.; Binder, K.; Paul, W. The Single Polymer Chain Phase Diagram: New Insights from a New Sim- ulation Method.J. Polym. Sci., Part B: Polym. Phys. 2006,44, 2542-2555

    work page 2006

  33. [33]

    First-Order Phase Transition in a Stiff Polymer Chain.Chem

    Noguchi, H.; Yoshikawa, K. First-Order Phase Transition in a Stiff Polymer Chain.Chem. Phys. Lett.1997,278, 184-188

    work page 1997

  34. [34]

    Phase Diagram of a Semiflexible Polymer Chain in aθSolvent: Application to Protein Folding.J

    Doniach, S.; Garel, T.; Orland, H. Phase Diagram of a Semiflexible Polymer Chain in aθSolvent: Application to Protein Folding.J. Chem. Phys.1996,105, 1601

    work page 1996

  35. [35]

    Phase Transitions of Single Semistiff Polymer Chains.J

    Bastolla, U.; Grassberger, P. Phase Transitions of Single Semistiff Polymer Chains.J. Stat. Phys.1997,89, 1061- 1078. 12

    work page 1997

  36. [36]

    Phase Transitions of Sin- gle Polymer Chains and of Polymer Solutions: Insights from Monte Carlo Simulations.J

    Binder, K.; Paul, W.; Strauch, T.; Rampf, F.; Ivanov, V.; Luettmer-Strathmann, J. Phase Transitions of Sin- gle Polymer Chains and of Polymer Solutions: Insights from Monte Carlo Simulations.J. Phys.: Condens. Mat- ter2008,20, 494215

    work page

  37. [37]

    P.; Paul, W.; Binder, K

    Taylor, M. P.; Paul, W.; Binder, K. Phase Transitions of a Single Polymer Chain: A Wang-Landau Simulation Study.J. Chem. Phys.2009,131, 114907

    work page 2009

  38. [38]

    P.; Paul, W.; Binder, K

    Taylor, M. P.; Paul, W.; Binder, K. Applications of the Wang-Landau Algorithm to Phase Transitions of a Single Polymer Chain.Polymer Sci. Ser. C2013,55, 23-38

    work page

  39. [39]

    L.; Oster, L.; Sportsman, R.; Phillips, M.; Knobler, C

    Duran-Meza, A. L.; Oster, L.; Sportsman, R.; Phillips, M.; Knobler, C. M.; Gelbart, W. M. Long ssRNA Under- goes Continuous Compaction in the Presence of Polyva- lent Cations.Biophys. J.2023,122, 3469-3475

    work page 2023

  40. [40]

    General Criterion for Controllable Conformational Transitions of Single- and Double-Stranded DNA

    Diamant, H.; Andelman, D. General Criterion for Controllable Conformational Transitions of Single- and Double-Stranded DNA. Unpublished,2001. Preprint available at https://doi.org/10.48550/arXiv.cond- mat/0109253

    work page doi:10.48550/arxiv.cond- 2001

  41. [41]

    Pruned-Enriched Rosenbluth Method: Simulations ofθPolymers of Chain Length up to 1 000 000.Phys

    Grassberger, P. Pruned-Enriched Rosenbluth Method: Simulations ofθPolymers of Chain Length up to 1 000 000.Phys. Rev. E1997,56, 3682

    work page

  42. [42]

    N.; Rosenbluth, A

    Rosenbluth, M. N.; Rosenbluth, A. W. Monte Carlo Cal- culation of the Average Extension of Molecular Chains. J. Chem. Phys.1955,23, 356

    work page 1955

  43. [43]

    T.; Erpenbeck, J

    Wall, F. T.; Erpenbeck, J. J. New Method for the Sta- tistical Computation of Polymer Dimensions.J. Chem. Phys.1959,30, 634

    work page 1959

  44. [44]

    P.; Grassberger, P

    Hsu, H. P.; Grassberger, P. A Review of Monte Carlo Simulations of Polymers with PERM.J. Stat. Phys. 2011,144, 597-637

    work page 2011

  45. [45]

    Golestanian, R.; Kardar, M.; Liverpool, T. B. Collapse of Stiff Polyelectrolytes due to Counterion Fluctuations. Phys. Rev. Lett.1999,82, 4456

    work page 1999

  46. [46]

    Binding Bacteria to Highly Branched Poly(N-Isopropyl Acrylamide) Modified with Vancomycin Induces the Coil-to-Globule Transition

    Shepherd, J.; Sarker, P.; Swindells, K.; Douglas, I.; Mac- Neil, S.; Swanson, L.; Rimmer, S. Binding Bacteria to Highly Branched Poly(N-Isopropyl Acrylamide) Modified with Vancomycin Induces the Coil-to-Globule Transition. J. Am. Chem. Soc.2010,132, 1736-1737

    work page 2010

  47. [47]

    Patel, A.; Malinovska, L.; Saha, S.; Wang, J.; Alberti, S.; Krishnan, Y.; Hyman, A. A. ATP as a Biological Hydrotrope.Science2017,356, 753-756

    work page

  48. [48]

    M.; Gavrilov, A

    Rumyantsev, A. M.; Gavrilov, A. A.; Johner, A. Com- plete Diagram of Conformational Regimes for Polyam- pholytic Disordered Proteins.Macromolecules2024,57, 5533-5544

    work page

  49. [49]

    de Gennes, P.-G.Scaling Concepts in Polymer Physics; Cornell University, Ithaca,1979. 13 for Table of Contents use only Collapse of a single polymer chain: Effects of chain stiffness and attraction range Persistence length 𝑙𝑝 Attraction range 𝑟𝑐 hairpin No transition Promote collapse Suppress collapse Abrupt transition Gradual transition

    work page 1979