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Mechanical property of the helical configuration for a twisted intrinsically straight biopolymer

  • Zicong ZhouEmail author
  • Chen-Xu Wu
Original Article
  • 21 Downloads

Abstract

We explore the effects of two typical torques on the mechanical property of the helical configuration for an intrinsically straight filament or biopolymer either in three-dimensional space or on a cylinder. One torque is parallel to the direction of a uniaxial applied force, and is coupled to the cross section of the filament. We obtain some algebraic equations for the helical configuration and find that the boundary conditions are crucial. In three-dimensional space, we show that the extension is always a monotonic function of the applied force. On the other hand, for a filament confined on a cylinder, the twisting rigidity and torque coupled to the cross section are irrelevant in forming a helix if the filament is isotropic and under free boundary condition. However, the twisting rigidity and the torque coupled to the cross section become crucial when the Euler angle at two ends of the filament are fixed. Particularly, the extension of a helix can subject to a first-order transition so that in such a condition a biopolymer can act as a switch or sensor in some biological processes. We also present several phase diagrams to provide the conditions to form a helix.

Keywords

Mechanical property Twisted biopolymer Helix Phase transition 

Notes

Acknowledgements

This work has been supported by the MOST of China. Funding was provided by National Natural Science Foundation of China (Grant no. 11574256).

References

  1. Allard JF, Rutenberg AD (2009) Pulling helices inside bacteria: imperfect helices and rings. Phys Rev Lett 102:158105CrossRefGoogle Scholar
  2. Andrews SS, Arkin AP (2007) A mechanical explanation for cytoskeletal rings and helices in bacteria. Biophys J 93:1872–1884CrossRefGoogle Scholar
  3. Benham CJ (1977) Elastic model of supercoiling. Proc Natl Acad Sci USA 74:2397–2401CrossRefGoogle Scholar
  4. Benham CJ (1989) Onset of writhing in circular elastic polymers. Phys Rev A 39:2582CrossRefGoogle Scholar
  5. Bustamante C, Marko JF, Siggia ED, Smith S (1994) Entropic elasticity of lambda-phage DNA. Science 265:1599–1600CrossRefGoogle Scholar
  6. Carballido-López R (2006) The bacterial actin-like cytoskeleton. Microbiol Mol Biol Rev 70:888–909CrossRefGoogle Scholar
  7. Chouaieb N, Goriely A, Maddocks JH (2006) Helices. Proc Natl Acad Sci USA 103:9398–9403CrossRefGoogle Scholar
  8. Daniel RA, Errington J (2003) Control of cell morphogenesis in bacteria: two distinct ways to make a rod-shaped cell. Cell 113:767–776CrossRefGoogle Scholar
  9. Erickson HP, Taylor DW, Taylor KA, Bramhill D (1996) Bacterial cell division protein FtsZ assembles into protofilament sheets and minirings, structural homologs of tubulin polymers. Proc Natl Acad Sci USA 93:519CrossRefGoogle Scholar
  10. Esue O, Wirtz D, Tseng Y (2006) GTPase activity, structure, and mechanical properties of filaments assembled from bacterial cytoskeleton protein MreB. J Bacteriol 188:968–976CrossRefGoogle Scholar
  11. Fain B, Rudnick J (1999) Conformations of closed DNA. Phys Rev E 60:7239CrossRefGoogle Scholar
  12. Fain B, Rudnick J, Östlund S (1997) Conformations of linear DNA. Phys Rev E 55:7364CrossRefGoogle Scholar
  13. Gitai Z, Dye N, Shapiro L (2004) An actin-like gene can determine cell polarity in bacteria. Proc Natl Acad Sci USA 101:8643–8648CrossRefGoogle Scholar
  14. Gitai Z, Dye N, Reisenauer A, Wachi M, Shapiro L (2005) MreB actin-mediated segregation of a specific region of a bacterial chromosome. Cell 120:329–341CrossRefGoogle Scholar
  15. Goldstein H (2002) Classical mechanics, 3rd edn. Addison-Wesley, BostonGoogle Scholar
  16. Goriely A, Shipman P (2000) Dynamics of helical strips. Phys Rev E 6:4508CrossRefGoogle Scholar
  17. Iwai N, Nagai K, Wachi M (2002) Novel S-benzylisothiourea compound that induces spherical cells in Escherichia coli probably by acting on a rod-shape-determining protein(s) other than penicillin-binding protein 2. Biosci Biotechnol Biochem 66:2658–2662CrossRefGoogle Scholar
  18. Jones LJF, Carballido-López R, Errington J (2001) Control of cell shape in bacteria: helical, actin-like filaments in Bacillus subtilis. Cell 104:913–922CrossRefGoogle Scholar
  19. Jung Y, Ha BY (2019) Confinement induces helical organization of chromosome-like polymers. Sci Rep 9:869Google Scholar
  20. Kessler DA, Rabin Y (2003) Stretching instability of helical springs. Phys Rev Lett 90:024301CrossRefGoogle Scholar
  21. Kratky O, Porod G (1949) Rontgenuntersuchung gelöster Fadenmoleküle. Recl Trav Chim Pays-Bas 68:1106CrossRefGoogle Scholar
  22. Kruse T, Bork-Jensen J, Gerdes K (2005) The morphogenetic MreBCD proteins of Escherichia coli form an essential membrane-bound complex. Mol Microbiol 55:78–89CrossRefGoogle Scholar
  23. Love AEH (1944) A treatise on the mathematical theory of elasticity. Dover, New YorkGoogle Scholar
  24. Marko JF, Siggia ED (1994) Fluctuations and supercoiling of DNA. Science 265:506–508CrossRefGoogle Scholar
  25. Marko JF, Siggia ED (1998) Stretching DNA. Macromolecules 28:8759–8770CrossRefGoogle Scholar
  26. Moukhtar J, Fontaine E, Faivre-Moskalenko C, Arnéodo A (2007) Probing persistence in DNA curvature properties with atomic force microscopy. Phys Rev Lett 98:178101CrossRefGoogle Scholar
  27. Panyukov SV, Rabin Y (2000) Thermal fluctuations of elastic filaments with spontaneous curvature and torsion. Phys Rev Lett 85:2404CrossRefGoogle Scholar
  28. Panyukov SV, Rabin Y (2001) Fluctuating elastic rings: statics and dynamics. Phys Rev E 64:011909CrossRefGoogle Scholar
  29. Panyukov SV, Rabin Y (2002) On the deformation of fluctuating chiral ribbons. Europhys Lett 57:512–518CrossRefGoogle Scholar
  30. Russell JH, Keiler KC (2007) Peptide signals encode protein localization. J Bacteriol 189:7581–7585CrossRefGoogle Scholar
  31. Shih Y-L, Le T, Rothfield LI (2003) Division site selection in Escherichia coli involves dynamic redistribution of Min proteins within coiled structures that extend between the two cell poles. Proc Natl Acad Sci USA 100:7865–7870CrossRefGoogle Scholar
  32. Smith B, Zastavker YV, Benedek GB (2001) Tension-induced straightening transition of self-assembled helical ribbons. Phys Rev Lett 87:278101CrossRefGoogle Scholar
  33. Srinivasan R, Mishra M, Murata-Hori M, Balasubramanian MK (2007) Filament formation of the Escherichia coli actin-related protein, MreB, in fission yeast. Curr Biol 17:266–272CrossRefGoogle Scholar
  34. Srinivasan R, Mishra M, Wu L, Yin Z, Balasubramanian MK (2008) The bacterial cell division protein FtsZ assembles into cytoplasmic rings in fission yeast. Genes Dev 22:1741–1746CrossRefGoogle Scholar
  35. Starostin EL, van der Heijden GHM (2010) Tension-induced multistability in inextensible helical ribbons. Phys Rev Lett 101:084301CrossRefGoogle Scholar
  36. Taghbalout A, Rothfield L (2007) RNaseE and the other constituents of the RNA degradosome are components of the bacterial cytoskeleton. Proc Natl Acad Sci USA 104:1667–1672CrossRefGoogle Scholar
  37. Tanaka F, Takahashi H (1985) Elastic theory of supercoiled DNA. J Chem Phys 83:6017CrossRefGoogle Scholar
  38. Thanedar S, Margolin W (2004) FtsZ exhibits rapid movement and oscillation waves in helix-like patterns in Escherichia coli. Curr Biol 14:1167–1173CrossRefGoogle Scholar
  39. Vaillant C, Audit B, Arnéodo A (2005) Thermodynamics of DNA loops with long-range correlated structural disorder. Phys Rev Lett 95:068101CrossRefGoogle Scholar
  40. van der Heijden GHM (2001) The static deformation of a twisted elastic rod constrained to lie on a cylinder. Proc R Soc A 457:695–715CrossRefGoogle Scholar
  41. Vats P, Rothfield L (2007) Duplication and segregation of the actin (MreB) cytoskeleton during the prokaryotic cell cycle. Proc Natl Acad Sci USA 104:17795–17800CrossRefGoogle Scholar
  42. Zhou Z (2007) Elasticity of two-dimensional filaments with constant spontaneous curvature. Phys Rev E 76:061913CrossRefGoogle Scholar
  43. Zhou Z (2018) Novel relationships between some coordinate systems and their effects on mechanics of an intrinsically curved filament. J Phys Commun 2:035008CrossRefGoogle Scholar
  44. Zhou Z, Lai P-Y, Joós B (2005) Elasticity and stability of a helical filament. Phys Rev E 71:052801CrossRefGoogle Scholar
  45. Zhou Z, Joós B, Lai P-Y, Young Y-S, Jan J-H (2007) Elasticity and stability of a helical filament with spontaneous curvatures and isotropic bending rigidity. Mod. Phys. Lett. B 21:1895CrossRefGoogle Scholar
  46. Zhou Z, Lin F-T, Hung C-Y, Wu H-Y, Chen B-H (2014) Curvature induced discontinuous transition for semiflexible biopolymers. J Phys Soc Jpn 83:044802CrossRefGoogle Scholar
  47. Zhou Z, Joós B, Wu C-X (2017) Stability of the helical configuration of an intrinsically straight semiflexible biopolymer inside a cylindrical cell. AIP Adv 7:125003CrossRefGoogle Scholar

Copyright information

© European Biophysical Societies' Association 2019

Authors and Affiliations

  1. 1.Department of PhysicsTamkang UniversityNew Taipei CityChina
  2. 2.Department of Physics and ITPAXiamen UniversityXiamenChina

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