Skip to main content
SpringerLink
Log in

Impact of flexibility on the aggregation of polymeric macromolecules

  • Regular Article - Soft Matter
  • Published:
The European Physical Journal E Aims and scope Submit manuscript

Abstract

Dependence of the dimerization probability and the aggregation behavior of polymeric macromolecules on their flexibility is studied using Langevin dynamics simulations. It is found that the dimerization probability is a non-monotonic function of the polymers persistence length. For a given value of inter-polymer attraction strength, semiflexible polymers have lower dimerization probability relative to flexible and rigid polymers of the same length. The threshold temperature of the formation of aggregates in a many-polymer system and its dependence on the polymers persistence length is also investigated. The simulation results of two- and many-polymer systems are in good agreement and show how the amount of flexibility affects the dimerization and the aggregation behaviors of polymeric macromolecules.

Graphical Abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. G.M. Whitesides, J.P. Mathias, C.T. Seto, Molecular self-assembly and nanochemistry: a chemical strategy for the synthesis of nanostructures. Science 254, 1312–1319 (1991). https://doi.org/10.1126/science.1962191

    Article  ADS  Google Scholar 

  2. G.M. Whitesides, B. Grzybowski, Self-assembly at all scales. Science 295, 2418–2421 (2002). https://doi.org/10.1126/science.1070821

    Article  ADS  Google Scholar 

  3. G.M. Whitesides, M. Boncheva, Beyond molecules: self-assembly of mesoscopic and macroscopic components. Proc. Natl. Acad. Sci. U.S.A. 99, 4769–4774 (2002). https://doi.org/10.1073/pnas.082065899

    Article  ADS  Google Scholar 

  4. M. Rubenstein, A. Cornejo, R. Nagpal, Programmable self-assembly in a thousand-robot swarm. Science 345, 795–799 (2014). https://doi.org/10.1126/science.1254295

    Article  ADS  Google Scholar 

  5. J. Wang, K. Liu, R. Xing, X. Yan, Peptide self-assembly: thermodynamics and kinetics. Chem. Soc. Rev. 45, 5589–5604 (2016). https://doi.org/10.1039/C6CS00176A

    Article  Google Scholar 

  6. S. Zhang, D.M. Marini, W. Hwang, S. Santoso, Design of nanostructured biological materials through self-assembly of peptides and proteins. Curr. Opin. Chem. Biol. 6, 865–871 (2002). https://doi.org/10.1016/S1367-5931(02)00391-5

    Article  Google Scholar 

  7. K. Rajagopal, J.P. Schneider, Self-assembling peptides and proteins for nanotechnological applications. Curr. Opin. Struct. Biol. 14, 480–486 (2004). https://doi.org/10.1016/j.sbi.2004.06.006

    Article  Google Scholar 

  8. E. Gazit, Self-assembled peptide nanostructures: the design of molecular building blocks and their technological utilization. Chem. Soc. Rev. 36, 1263–1269 (2007). https://doi.org/10.1039/b605536m

    Article  Google Scholar 

  9. X. Zhao, S. Zhang, Molecular designer self-assembling peptides. Chem. Soc. Rev. 35, 1105–1110 (2006). https://doi.org/10.1039/b511336a

    Article  Google Scholar 

  10. G. Chen, G.J. Mohamed, Complex protein patterns formation via salt-induced self-assembly and droplet evaporation. Eur. Phys. J. E 33, 19–26 (2010). https://doi.org/10.1140/epje/i2010-10649-4

    Article  Google Scholar 

  11. C. Tang, R.V. Ulijn, A. Saiani, Self-assembly and gelation properties of glycine/leucine Fmoc-dipeptides. Eur. Phys. J. E 36, 111 (2013). https://doi.org/10.1140/epje/i2013-13111-3

    Article  Google Scholar 

  12. Q. Zhang, M. Li, C. Zhu, G. Nurumbetov, Z. Li, P. Wilson, K. Kempe, D.M. Haddleton, Well-defined protein/peptide-polymer conjugates by aqueous Cu-LRP: synthesis and controlled self-assembly. J. Am. Chem. Soc. 137, 9344–9353 (2015). https://doi.org/10.1021/jacs.5b04139

    Article  Google Scholar 

  13. C.J.C. Edwards-Gayle, I.W. Hamley, Self-assembly of bioactive peptides, peptide conjugates, and peptide mimetic materials. Org. Biomol. Chem. 15, 5867–5876 (2017). https://doi.org/10.1039/c7ob01092c

    Article  Google Scholar 

  14. Z. Yang, H. Xu, X. Zhao, Designer self-assembling peptide hydrogels to engineer 3d cell microenvironments for cell constructs formation and precise oncology remodeling in ovarian cancer. Adv. Sci. 7, 1903718 (2020). https://doi.org/10.1002/advs.201903718

    Article  Google Scholar 

  15. J. Wang, J.Q.M. Choi, A.S. Holehouse, H.O. Lee, X. Zhang, M. Jahnel, S. Maharana, R. Lemaitre, A. Pozniakovsky, D. Drechsel, I. Poser, R.V. Pappu, S. Alberti, A.A. Hyman, A molecular grammar governing the driving forces for phase separation of prion-like rna binding proteins. Cell 174, 688–699 (2018). https://doi.org/10.1016/j.cell.2018.06.006

    Article  Google Scholar 

  16. A.A. Hyman, C.A. Weber, F. Jülicher, Liquid-liquid phase separation in biology. Annu. Rev. Cell Dev. Biol. 30, 39–58 (2014). https://doi.org/10.1146/annurev-cellbio-100913-013325

    Article  Google Scholar 

  17. C.P. Brangwynne, C.R. Eckmann, D.S. Courson, A. Rybarska, C. Hoege, J. Gharakhani, F. Jülicher, A.A. Hyman, Germline p granules are liquid droplets that localize by controlled dissolution/condensation. Science 324, 1729–1732 (2009). https://doi.org/10.1126/science.1172046

    Article  ADS  Google Scholar 

  18. Y. Lin, D.S.W. Protter, M.K. Rosen, R. Parker, Formation and maturation of phase-separated liquid droplets by rna-binding proteins. Mol. Cell 60, 208–219 (2015). https://doi.org/10.1016/j.molcel.2015.08.018

    Article  Google Scholar 

  19. T. Murakami, S. Qamar, J.Q. Lin, G.S.K. Schierle, E. Rees, A. Miyashita, A.R. Costa, R.B. Dodd, F.T.S. Chan, C.H. Michel, D. Kronenberg-Versteeg, Y. Li, S.Q.P. Yang, Y. Wakutani, W. Meadows, R.R. Ferry, L. Dong, G.G. Tartaglia, G. Favrin, W.Q.L. Lin, D.W. Dickson, M. Zhen, D. Ron, G. Schmitt-Ulms, P.E. Fraser, N.A. Shneider, C. Holt, M. Vendruscolo, C.F. Kaminski, P.S. George-Hyslop, Als/ftd mutation-induced phase transition of fus liquid droplets and reversible hydrogels into irreversible hydrogels impairs rnp granule function. Neuron 88, 678–690 (2015). https://doi.org/10.1016/j.neuron.2015.10.030

    Article  Google Scholar 

  20. D.T. Murray, M. Kato, Y. Lin, K.R. Thurber, I. Hung, S.L. McKnight, R. Tycko, Structure of fus protein fibrils and its relevance to self-assembly and phase separation of low-complexity domains. Cell 171, 615–627 (2017). https://doi.org/10.1016/j.cell.2017.08.048

    Article  Google Scholar 

  21. A. Patel, H.O. Lee, L. Jawerth, S. Maharana, M. Jahnel, M.Y. Hein, S. Stoynov, J. Mahamid, S. Saha, T.M. Franzmann, A. Pozniakovski, I. Poser, N. Maghelli, L.A. Royer, M. Weigert, E.W. Myers, S. Grill, D. Drechsel, A.A. Hyman, S. Alberti, A liquid-to-solid phase transition of the als protein fus accelerated by disease mutation. Cell 162, 1066–1077 (2015). https://doi.org/10.1016/j.cell.2015.07.047

    Article  Google Scholar 

  22. S. Boeynaems, S. Alberti, N.L. Fawzi, T. Mittag, M. Polymenidou, F. Rousseau, J. Schymkowitz, J. Shorter, B. Wolozin, L.V.D. Bosch, P. Tompa, M. Fuxreiter, Protein phase separation: a new phase in cell biology. Trends Cell Biol. 28, 420–435 (2018). https://doi.org/10.1016/j.tcb.2018.02.004

    Article  Google Scholar 

  23. A. Molliex, J. Temirov, J. Lee, M. Coughlin, A.P. Kanagaraj, H.J. Kim, T. Mittag, J.P. Taylor, Phase separation by low complexity domains promotes stress granule assembly and drives pathological fibrillization. Cell 163, 123–133 (2015). https://doi.org/10.1016/j.cell.2015.09.015

    Article  Google Scholar 

  24. K.A. Burke, A.M. Janke, C.L. Rhine, N.L. Fawzi, Residue-by-residue view of in vitro fus granules that bind the c-terminal domain of rna polymerase ii. Mol. Cell 60, 231–241 (2015). https://doi.org/10.1016/j.molcel.2015.09.006

    Article  Google Scholar 

  25. Z. Feng, X. Chen, X. Wu, M. Zhang, Formation of biological condensates via phase separation: characteristics, analytical methods, and physiological implications. J. Biol. Chem. 294, 14823–14835 (2019). https://doi.org/10.1074/jbc.REV119.007895

    Article  Google Scholar 

  26. C.W. Pak, M. Kosno, A.S. Holehouse, S.B. Padrick, A. Mittal, R. Ali, A.A. Yunus, D.R. Liu, R.V. Pappu, M.K. Rosen, Sequence determinants of intracellular phase separation by complex coacervation of a disordered protein. Mol. Cell 63, 72–85 (2016). https://doi.org/10.1016/j.molcel.2016.05.042

    Article  Google Scholar 

  27. G.L. Dignon, W. Zheng, Y.C. Kim, J. Mittal, Temperature-controlled liquid-liquid phase separation of disordered proteins. ACS Cent. Sci. 5, 821–830 (2019). https://doi.org/10.1021/acscentsci.9b00102

    Article  Google Scholar 

  28. S. Alberti, D. Dormann, Liquid-liquid phase separation in disease. Annu. Rev. Genet. 53, 171–194 (2019). https://doi.org/10.1146/annurev-genet-112618-043527

    Article  Google Scholar 

  29. A. Peng, S.C. Weber, Evidence for and against liquid-liquid phase separation in the nucleus. Non-Coding RNA 5, 50 (2019). https://doi.org/10.3390/ncrna5040050

    Article  Google Scholar 

  30. F.G. Quiroz, N.K. Li, S. Roberts, P. Weber, M. Dzuricky, I. Weitzhandler, Y.G. Yingling, A. Chilkoti, Intrinsically disordered proteins access a range of hysteretic phase separation behaviors. Sci. Adv. 5, 5177 (2019). https://doi.org/10.1126/sciadv.aax5177

    Article  ADS  Google Scholar 

  31. B.S. Schuster, E.H. Reed, R. Parthasarathy, C.N. Jahnke, R.M. Caldwell, J.G. Bermudez, H. Ramage, M.C. Good, D.A. Hammer, Controllable protein phase separation and modular recruitment to form responsive membraneless organelles. Nat. Commun. 9, 2985 (2018). https://doi.org/10.1038/s41467-018-05403-1

    Article  ADS  Google Scholar 

  32. I.E. Vega, A. Umstead, N.M. Kanaan, Efhd2 affects tau liquid-liquid phase separation. Front. Neurosci. 13, 845 (2019). https://doi.org/10.3389/fnins.2019.00845

    Article  Google Scholar 

  33. Q. Li, X. Peng, Y. Li, W. Tang, J. Zhu, J. Huang, Y. Qi, Z. Zhang, Llpsdb: A database of proteins undergoing liquid-liquid phase separation In Vitro. Nucleic Acids Res. (2019). https://doi.org/10.1093/nar/gkz778

    Article  Google Scholar 

  34. O. Adame-Arana, C.A. Weber, V. Zaburdaev, J. Prost, F. Jülicher, Liquid phase separation controlled by ph. Biophys. J . 119, 1590–1605 (2020). https://doi.org/10.1016/j.bpj.2020.07.044

    Article  Google Scholar 

  35. G. Krainer, T.J. Welsh, J.A. Joseph, J.R. Espinosa, S. Wittmann, E. Csilléry, A. Sridhar, Z. Toprakcioglu, G. Gudiškytė1, M.A. Czekalska, W.E. Arter, J. Guillén-Boixet, T.M. Franzmann, S. Qamar, P.S. George-Hyslop, A.A. Hyman, R. Collepardo-Guevara, S. Alberti, T.P.J. Knowles, Reentrant liquid condensate phase of proteins is stabilized by hydrophobic and non-ionic interactions. Nat. Commun. 12, 1085 (2021). https://doi.org/10.1038/s41467-021-21181-9

  36. S. Zhang, L. Yan, M. Altman, M. Lässle, H. Nugent, F. Frankel, D.A. Lauffenburger, G.M. Whitesides, A. Rich, Biological surface engineering: a simple system for cell pattern formation. Biomaterials 20, 1213–1220 (1999). https://doi.org/10.1016/S0142-9612(99)00014-9

    Article  Google Scholar 

  37. Y.Q.P. Jiao, F.Q.Z. Cui, Surface modification of polyester biomaterials for tissue engineering. Biomed. Mater. 2, 24–37 (2007). https://doi.org/10.1088/1748-6041/2/4/R02

    Article  ADS  Google Scholar 

  38. S. Nir, D. Zanuy, T. Zada, O. Agazani, C. Aleman, D.E. Shalev, M. Reches, Tailoring the self-assembly of a tripeptide for the formation of antimicrobial surfaces. Nanoscale 11, 8752–8759 (2019). https://doi.org/10.1039/C8NR10043H

    Article  Google Scholar 

  39. S. Zhang, T.C. Holmes, C.M. DiPersio, R.O. Hynes, X. Su, A. Rich, Self-complementary oligopeptide matrices support mammalian cell attachment. Biomaterials 16, 1385–1393 (1995). https://doi.org/10.1016/0142-9612(95)96874-Y

    Article  Google Scholar 

  40. E.C. Wu, S. Zhang, C.A.E. Hauser, Self-assembling peptides as cell-interactive scaffolds. Adv. Funct. Mater. 22, 456–468 (2012). https://doi.org/10.1002/adfm.201101905

    Article  Google Scholar 

  41. L. Sun, C. Zheng, T.J. Webster, Self-assembled peptide nanomaterials for biomedical applications: promises and pitfalls. Int. J. Nanomedicine 12, 73–86 (2017). https://doi.org/10.2147/IJN.S117501

    Article  Google Scholar 

  42. S. Zhang, Fabrication of novel biomaterials through molecular self-assembly. Nat. Biotechnol. 21, 1171–1178 (2003). https://doi.org/10.1038/nbt874

    Article  Google Scholar 

  43. L. Adler-Abramovich, E. Gazit, The physical properties of supramolecular peptide assemblies: from building block association to technological applications. Chem. Soc. Rev. 43, 6881–6893 (2014). https://doi.org/10.1039/c4cs00164h

    Article  Google Scholar 

  44. D.M. Shapiro, G. Mandava, S.E. Yalcin, P. Arranz-Gibert, P.J. Dah, C. Shipps, Y. Gu, V. Srikanth, A.I. Salazar-Morales, J.P. ÓBrien, K. Vanderschuren, D. Vu, V.S. Batista, N.S. Malvankar, F.J. Isaacs, Protein nanowires with tunable functionality and programmable self-assembly using sequence-controlled synthesis. Nat. Commun. 13, 829 (2022). https://doi.org/10.1038/s41467-022-28206-x

    Article  ADS  Google Scholar 

  45. R.W. Carrell, B. Gooptu, Conformational changes and disease-serpins, prions and Alzheimer’s. Curr. Opin. Struct. Biol. 8, 799–809 (1998). https://doi.org/10.1016/S0959-440X(98)80101-2

    Article  Google Scholar 

  46. C. Soto, G.P. Saborío, Prions: Disease propagation and disease therapy by conformational transmission. Trends Mol. Med. 7, 109–114 (2001). https://doi.org/10.1016/S1471-4914(01)01931-1

    Article  Google Scholar 

  47. S.T. Ferreira, F.G. De Felice, Protein dynamics, folding and misfolding: from basic physical chemistry to human conformational diseases. FEBS Lett. 498, 129–134 (2001). https://doi.org/10.1016/S0014-5793(01)02491-7

    Article  Google Scholar 

  48. E. Žerovnik, Amyloid-fibril formation: proposed mechanisms and relevance to conformational disease. Eur. J. Biochem. 269, 3362–3371 (2002). https://doi.org/10.1046/j.1432-1033.2002.03024.x

    Article  Google Scholar 

  49. N. Ghoshal, F. García-Sierra, J. Wuu, S. Leurgans, D.A. Bennett, R.W. Berry, L.I. Binder, Tau conformational changes correspond to impairments of episodic memory in mild cognitive impairment and Alzheimer’s disease. Exp. Neurol. 177, 475–493 (2002). https://doi.org/10.1006/exnr.2002.8014

    Article  Google Scholar 

  50. E. Gazit, Mechanisms of amyloid fibril self-assembly and inhibition: model short peptides as a key research tool. FEBS J. 272, 5971–5978 (2005). https://doi.org/10.1111/j.1742-4658.2005.05022.x

    Article  Google Scholar 

  51. S. Lee, T.H.T. Trinh, M. Yoo, J. Shin, H. Lee, J. Kim, E. Hwang, Y.-B. Lim, C. Ryou, Self-assembling peptides and their application in the treatment of diseases. Int. J. Mol. Sci. 20, 5850 (2019). https://doi.org/10.3390/ijms20235850

    Article  Google Scholar 

  52. H. Yang, S.Q.Y. Fung, M. Pritzker, P. Chen, Modification of hydrophilic and hydrophobic surfaces using an ionic-complementary peptide. PLoS ONE 2, 1325 (2007). https://doi.org/10.1371/journal.pone.0001325

    Article  ADS  Google Scholar 

  53. M. Stefani, Protein folding and misfolding on surfaces. Int. J. Mol. Sci. 9, 2515–2542 (2008). https://doi.org/10.3390/ijms9122515

    Article  Google Scholar 

  54. S. Jun, Y. Hong, H. Imamura, B.Q.Y. Ha, J. Bechhoefer, P. Chen, Self-assembly of the ionic peptide EAK16: the effect of charge distributions on self-assembly. Biophys. J . 87, 1249–1259 (2004). https://doi.org/10.1529/biophysj.103.038166

    Article  Google Scholar 

  55. M. Reches, E. Gazit, Controlled patterning of aligned self-assembled peptide nanotubes. Nat. Nanotechnol. 1, 195–200 (2006). https://doi.org/10.1038/nnano.2006.139

    Article  ADS  Google Scholar 

  56. Y.Q.C. Lin, E.J. Petersson, Z. Fakhraai, Surface effects mediate self-assembly of Amyloid-\(\beta \) peptides. ACS Nano 8, 10178–10186 (2014). https://doi.org/10.1021/nn5031669

    Article  Google Scholar 

  57. S. Emamyari, H. Fazli, pH-dependent self-assembly of EAK16 peptides in the presence of a hydrophobic surface: Coarse-grained molecular dynamics simulation. Soft Matter 10, 4248–4257 (2014). https://doi.org/10.1039/c4sm00307a

    Article  ADS  Google Scholar 

  58. S. Emamyari, F. Kargar, V. Sheikh-hasani, S. Emadi, H. Fazli, Mechanisms of the self-assembly of EAK16-family peptides into fibrillar and globular structures: molecular dynamics simulations from nano- to micro-seconds. Eur. Biophys. J. 44, 263–276 (2015). https://doi.org/10.1007/s00249-015-1024-y

    Article  Google Scholar 

  59. B. Yang, D.J. Adams, M. Marlow, M. Zelzer, Surface-mediated supramolecular self-assembly of protein, peptide, and nucleoside derivatives: from surface design to the underlying mechanism and tailored functions. Langmuir 34, 15109–15125 (2018). https://doi.org/10.1021/acs.langmuir.8b01165

    Article  Google Scholar 

  60. D. Wouters, U.S. Schubert, Nanolithography and nanochemistry: Probe-related patterning techniques and chemical modification for nanometer-sized devices. Angew. Chem. Int. Ed. 43, 2480–2495 (2004). https://doi.org/10.1002/anie.200300609

    Article  Google Scholar 

  61. C.J. Bowerman, B.L. Nilsson, Self-assembly of amphipathic \(\beta \)-sheet peptides: insights and applications. Biopolymers (Pept. Sci.) 98, 169–184 (2012). https://doi.org/10.1002/bip.22058

    Article  Google Scholar 

  62. M. Rubinstein, R.H. Colby, Polymer Physics, 1st edn. (Oxford University Press, New York, 2003)

    Google Scholar 

  63. F. Huang, W.M. Nau, A conformational flexibility scale for amino acids in peptides. Angew. Chem. Int. Ed. 42, 2269–2272 (2003). https://doi.org/10.1002/anie.200250684

    Article  Google Scholar 

  64. S. Rekhi, D.S. Devarajan, M.P. Howard, Y.C. Kim, A. Nikoubashman, J. Mittal, Role of strong localized vs weak distributed interactions in disordered protein phase separation. J. Phys. Chem. B 127(17), 3829–3838 (2023). https://doi.org/10.1021/acs.jpcb.3c00830

    Article  Google Scholar 

  65. F. Weik, R. Weeber, K. Szuttor, K. Breitsprecher, J. Graaf, M. Kuron, J. Landsgesell, H. Menke, D. Sean, C. Holm, Espresso 4.0 - an extensible software package for simulating soft matter systems. Eur. Phys. J.: Spec. Top. 227, 1789–1816 (2019). https://doi.org/10.1140/epjst/e2019-800186-9

    Article  Google Scholar 

  66. A.P. Thompson, H.M. Aktulga, R. Berger, D.S. Bolintineanu, W.M. Brown, P.S. Crozier, P.J. Veld, A. Kohlmeyer, S.G. Moore, T.D. Nguyen, R. Shan, M.J. Stevens, J. Tranchida, C. Trott, S.J. Plimpton, Lammps - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales. Comp. Phys. Comm. 271, 108171 (2022). https://doi.org/10.1016/j.cpc.2021.108171

    Article  MATH  Google Scholar 

  67. M.K. Singh, M. Hu, Y. Cang, H.P. Hsu, H. Therien-Aubin, K. Koynov, G. Fytas, K. Landfester, K. Kremer, Glass transition of disentangled and entangled polymer melts: single-chain-nanoparticles approach. Macromolecules 53(17), 7312–7321 (2020). https://doi.org/10.1021/acs.macromol.0c00550

    Article  ADS  Google Scholar 

  68. K.S. Silmore, M.P. Howard, A.Z. Panagiotopoulos, Vapour-liquid phase equilibrium and surface tension of fully flexible lennard-jones chains. Mol. Phys. 115(3), 320–327 (2017). https://doi.org/10.1080/00268976.2016.1262075

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), Prof. Yousef Sobouti Blvd., Zanjan, 45137-66731, Iran

    Soheila Emamyari, Masoud Mirzaei, Davood Fazli & Hossein Fazli

  2. Department of Biological Sciences, Institute for Advanced Studies in Basic Sciences (IASBS), Prof. Yousef Sobouti Blvd., Zanjan, 45137-66731, Iran

    Sarah Mohammadinejad & Hossein Fazli

Authors
  1. Soheila Emamyari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Masoud Mirzaei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sarah Mohammadinejad
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Davood Fazli
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Hossein Fazli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hossein Fazli.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Emamyari, S., Mirzaei, M., Mohammadinejad, S. et al. Impact of flexibility on the aggregation of polymeric macromolecules. Eur. Phys. J. E 46, 66 (2023). https://doi.org/10.1140/epje/s10189-023-00324-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epje/s10189-023-00324-4

Search

Navigation