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Optimal Kinematics of a Looped Filament

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Abstract

New kinematics of supercoiling of closed filaments as solutions of the elastic energy minimization are proposed. The analysis is based on the thin rod approximation of the linear elastic theory, under conservation of the self-linking number. The elastic energy is evaluated by means of bending contribution and torsional influence. Time evolution functions are described by means of piecewise polynomial transformations based on cubic spline functions. In contrast with traditional interpolation, the parameters, which define the cubic splines representing the evolution functions, are considered as the unknowns in a nonlinear optimization problem. We show how the coiling process is associated with conversion of mean twist energy into bending energy through the passage by an inflexional configuration in relation to geometric characteristics of the filament evolution. These results provide new insights on the folding mechanism and associated energy contents and may find useful applications in folding of macromolecules and DNA packing in cell biology.

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References

  1. Bauer, W.R., Crick, F.H.C., White, J.H.: Supercoiled DNA. Sci. Am. 243, 100–113 (1980)

    Google Scholar 

  2. Stasiak, A., Circular, D.NA.: In: Semlyen, J.A. (ed.) Large Ring Molecules, pp. 43–97. Wiley, New York (1996)

    Google Scholar 

  3. Cozzarelli, N.R., Wang, J.C. (eds.): DNA Topology and Its Biological Effects. Cold Spring Harbor Laboratory Press, New York (1990)

    Google Scholar 

  4. Demurtas, D., Amzallag, A., Rawdon, E.J., Maddocks, J.H., Dubochet, J., Stasiak, A.: Bending modes of DNA directly addressed by cryo-electron microscopy of DNA minicircles. Nucleic Acids Res. 37(9), 2882–2893 (2009)

    Article  Google Scholar 

  5. Víglasky, V., Valle, F., Adamcík, J., Joab, I., Podhradsky, D., Dietler, G.: Anthracycline-dependent heat-induced transition from positive to negative supercoiled DNA. Electrophoresis 24(11), 1703–1711 (2003)

    Article  Google Scholar 

  6. Amzallag, A., Vaillant, C., Jacob, M., Unser, M., Bednar, J., Kahn, J.D., Dubochet, J., Stasiak, A., Maddocks, J.H.: 3D reconstruction and comparison of shapes of DNA minicircles observed by cryo-electron microscopy. Nucleic Acids Res. 34(18), e125 (2006)

    Article  Google Scholar 

  7. Wiggins, P.A., van der Heijden, T., Moreno-Herrero, F., Spakowitz, A., Phillips, R., Widom, J., Dekker, C., Nelson, P.: High flexibility of DNA on short length scales probed by atomic force microscopy. Nat. Nanotechnol. 1, 137–141 (2006)

    Article  Google Scholar 

  8. Maggioni, F., Ricca, R.L.: Writhing and coiling of closed filaments. Proc. R. Soc. A 462, 3151–3166 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ricca, R.L., Maggioni, F.: Multiple folding and packing in DNA modeling. Comput. Math. Appl. 55, 1044–1053 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Michell, J.H.: On the stability of a bent and twisted wire. Messenger Math. 11, 181–184 (1989/1990)

    Google Scholar 

  11. Zajac, E.E.: Stability of two planar loop elasticas. J. Appl. Mech. 29, 136–142 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  12. Goriely, A.: Twisted elastic rings and the rediscoveries of Michell’s instability. J. Elast. 84, 281–299 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Benham, C.J.: Onset of writhing in circular elastic polymers. Phys. Rev. A 39, 2582–2586 (1989)

    Article  Google Scholar 

  14. LeBret, M.: Twist and writhing in short circular DNA according to first-order elasticity. Biopolymers 23, 1835–1867 (1984)

    Article  Google Scholar 

  15. Fuller, F.B.: The writhing number of a space curve. Proc. Natl. Acad. Sci. USA 68, 815–819 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  16. Coleman, B.D., Swigon, D.: Theory of supercoiled elastic rings with self-contact and its application to DNA plasmids. J. Elast. 60, 171–221 (2000)

    Article  MathSciNet  Google Scholar 

  17. Coleman, B.D., Swigon, D., Tobias, I.: Elastic stability of DNA configurations. II. Supercoiled plasmids with self-contact. Phys. Rev. E 61, 759–770 (2000)

    Article  MathSciNet  Google Scholar 

  18. Coleman, B.D., Tobias, I., Swigon, D.: Theory of influence of end conditions on self-contact in DNA loops. J. Chem. Phys. 103, 9101–9109 (1995)

    Article  Google Scholar 

  19. Schlick, T., Li, B., Olson, W.K.: The influence of salt on the structure and energetics of supercoiled DNA. Biophys. J. 67(6), 2146–2166 (1994)

    Article  Google Scholar 

  20. Timothy, P., Westcott, I.T., Olson, W.K.: Modeling self-contact forces in the elastic theory of DNA supercoiling. J. Chem. Phys. 107, 3967 (1997)

    Article  Google Scholar 

  21. Goriely, A., Tabor, M.: The nonlinear dynamics of filaments. Nonlinear Dyn. 21, 101–133 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Goriely, A., Nizette, M., Tabor, M.: On the dynamics of elastic strips. J. Nonlinear Sci. 11, 3–45 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Klapper, I.: Biological applications of the dynamics of twisted elastic rods. J. Comput. Phys. 125, 325–337 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Schlick, T.: Modeling superhelical DNA: recent analytical and dynamical approaches. Curr. Opin. Struct. Biol. 5, 245 (1995)

    Article  Google Scholar 

  25. Schlick, T., Olson, W.K.: Supercoiled DNA energetics and dynamics by computer simulation. J. Mol. Biol. 223, 1089–1119 (1992)

    Article  Google Scholar 

  26. Arganbright, D.E.: Practical Handbook of Spreadsheet Curves and Geometric Constructions. CRC Press, Boca Raton (1993)

    MATH  Google Scholar 

  27. Lockwood, E.H.: A Book of Curves. Cambridge University Press, Cambridge (1961)

    Book  MATH  Google Scholar 

  28. Kauffman, L.H.: Fourier Knots (1997). arXiv:q-alg/9711013v2

  29. Kamien, R.D.: The geometry of soft materials: a primer. Rev. Mod. Phys. 74, 953–971 (2002)

    Article  Google Scholar 

  30. Călugăreanu, G.: Sur les classes d’isotopie des nœuds tridimensionnels et leurs invariants. Czechoslov. Math. J. 11, 588–625 (1961)

    Google Scholar 

  31. White, J.H.: Self-linking and the Gauss integral in higher dimensions. Am. J. Math. 91, 693–728 (1969)

    Article  MATH  Google Scholar 

  32. Ricca, R.L.: The energy spectrum of a twisted flexible string under elastic relaxation. J. Phys. A, Math. Gen. 28, 2335–2352 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  33. Atkinson, K.E.: An Introduction to Numerical Analysis. Wiley, New York (1989)

    MATH  Google Scholar 

  34. de Boor, C.: A Practical Guide to Splines. Applied Mathematical Sciences, vol. 27. Springer, New York (1978)

    Book  MATH  Google Scholar 

  35. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research. Springer, New York (1999)

    Book  MATH  Google Scholar 

  36. Moffatt, H.K., Ricca, R.L.: Helicity and the Călugăreanu invariant. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 439, 411–429 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  37. Sumners, D.W.: Random knotting: theorems, simulations and applications. In: Ricca, R.L. (ed.) Lectures on Topological Fluid Mechanics. Lecture Notes in Mathematics, vol. 187, pp. 201–231. Springer, Berlin (2009)

    Google Scholar 

  38. Arsuaga, J., Tan, R.K.Z., Vazquez, M., Sumners, D.W., Harvey, S.C.: Investigation of viral DNA packaging using molecular mechanics models. Biophys. Chem. 101, 475–484 (2002)

    Article  Google Scholar 

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Acknowledgements

F. Maggioni would like to thank Renzo L. Ricca and David Swigon for discussions and helpful advices.

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Authors and Affiliations

  1. Department of Mathematics, Statistics, Computer Science and Applications, University of Bergamo, Via dei Caniana 2, 24127, Bergamo, Italy

    Francesca Maggioni & Marida Bertocchi

  2. Department of Mathematics & Statistics, University of Maryland, Baltimore, Baltimore County, MD, 21250, USA

    Florian A. Potra

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  1. Francesca Maggioni
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  2. Florian A. Potra
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Correspondence to Francesca Maggioni.

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Maggioni, F., Potra, F.A. & Bertocchi, M. Optimal Kinematics of a Looped Filament. J Optim Theory Appl 159, 489–506 (2013). https://doi.org/10.1007/s10957-013-0330-8

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  • DOI: https://doi.org/10.1007/s10957-013-0330-8

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