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.
Similar content being viewed by others
References
Bauer, W.R., Crick, F.H.C., White, J.H.: Supercoiled DNA. Sci. Am. 243, 100–113 (1980)
Stasiak, A., Circular, D.NA.: In: Semlyen, J.A. (ed.) Large Ring Molecules, pp. 43–97. Wiley, New York (1996)
Cozzarelli, N.R., Wang, J.C. (eds.): DNA Topology and Its Biological Effects. Cold Spring Harbor Laboratory Press, New York (1990)
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)
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)
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)
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)
Maggioni, F., Ricca, R.L.: Writhing and coiling of closed filaments. Proc. R. Soc. A 462, 3151–3166 (2006)
Ricca, R.L., Maggioni, F.: Multiple folding and packing in DNA modeling. Comput. Math. Appl. 55, 1044–1053 (2008)
Michell, J.H.: On the stability of a bent and twisted wire. Messenger Math. 11, 181–184 (1989/1990)
Zajac, E.E.: Stability of two planar loop elasticas. J. Appl. Mech. 29, 136–142 (1962)
Goriely, A.: Twisted elastic rings and the rediscoveries of Michell’s instability. J. Elast. 84, 281–299 (2006)
Benham, C.J.: Onset of writhing in circular elastic polymers. Phys. Rev. A 39, 2582–2586 (1989)
LeBret, M.: Twist and writhing in short circular DNA according to first-order elasticity. Biopolymers 23, 1835–1867 (1984)
Fuller, F.B.: The writhing number of a space curve. Proc. Natl. Acad. Sci. USA 68, 815–819 (1971)
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)
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)
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)
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)
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)
Goriely, A., Tabor, M.: The nonlinear dynamics of filaments. Nonlinear Dyn. 21, 101–133 (2000)
Goriely, A., Nizette, M., Tabor, M.: On the dynamics of elastic strips. J. Nonlinear Sci. 11, 3–45 (2001)
Klapper, I.: Biological applications of the dynamics of twisted elastic rods. J. Comput. Phys. 125, 325–337 (1996)
Schlick, T.: Modeling superhelical DNA: recent analytical and dynamical approaches. Curr. Opin. Struct. Biol. 5, 245 (1995)
Schlick, T., Olson, W.K.: Supercoiled DNA energetics and dynamics by computer simulation. J. Mol. Biol. 223, 1089–1119 (1992)
Arganbright, D.E.: Practical Handbook of Spreadsheet Curves and Geometric Constructions. CRC Press, Boca Raton (1993)
Lockwood, E.H.: A Book of Curves. Cambridge University Press, Cambridge (1961)
Kauffman, L.H.: Fourier Knots (1997). arXiv:q-alg/9711013v2
Kamien, R.D.: The geometry of soft materials: a primer. Rev. Mod. Phys. 74, 953–971 (2002)
Călugăreanu, G.: Sur les classes d’isotopie des nœuds tridimensionnels et leurs invariants. Czechoslov. Math. J. 11, 588–625 (1961)
White, J.H.: Self-linking and the Gauss integral in higher dimensions. Am. J. Math. 91, 693–728 (1969)
Ricca, R.L.: The energy spectrum of a twisted flexible string under elastic relaxation. J. Phys. A, Math. Gen. 28, 2335–2352 (1995)
Atkinson, K.E.: An Introduction to Numerical Analysis. Wiley, New York (1989)
de Boor, C.: A Practical Guide to Splines. Applied Mathematical Sciences, vol. 27. Springer, New York (1978)
Nocedal, J., Wright, S.J.: Numerical Optimization. Springer Series in Operations Research. Springer, New York (1999)
Moffatt, H.K., Ricca, R.L.: Helicity and the Călugăreanu invariant. Proc. R. Soc. Lond. Ser. A, Math. Phys. Sci. 439, 411–429 (1992)
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)
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)
Acknowledgements
F. Maggioni would like to thank Renzo L. Ricca and David Swigon for discussions and helpful advices.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-013-0330-8