Abstract
Motivated by the elastic rod model for DNA with intrinsic curvature, we study the solution space of the Euler-Lagrange equations governing isotropic, homogeneous, naturally curved Kirchhoff’s elastic thin rods. Our studies show that for each given total energy and twisting density, there are at most three solutions, aside from the case where the twisting density is some particular constant. We also propose in this paper a reasonable condition under which an improvement on the number of the solutions may be possible. Finally, numerical calculations are presented to support our conclusions.
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References
Bates, A.D., Maxwell, A.: DNA Topology. IRL Press, Oxford (1993)
Bauer, W.R., Lund, R.A., White, J.H.: Twist and writhe of a DNA loop containing intrinsic bends. Proc. Natl. Acad. Sci. USA 90, 833–837 (1993)
Benham, C.J.: Elastic model of supercoiling. Proc. Natl. Acad. Sci. USA 74, 2387–2401 (1977)
Benham, C.J.: Geometry and mechanics of DNA superhelicity. Biopolymers 22, 2477–2495 (1983)
Benham, C.J.: Theoretical analysis of conformational equilibria in superhelical DNA. Ann. Rev. Biophys. Biophys. Chem. 14, 23–45 (1985)
Benham, C.J.: The role of the stress resultant in determining mechanical equilibria of superhelical DNA. Biopolymers 26, 9–15 (1987)
Benham, C.J.: Onset of writhing in circular elastic polymers. Phys. Rev. A 39, 2582–2586 (1989)
Birkhoff, G., Rota, G.-C.: Ordinary Differential Equations, 2nd edn. Wiley, New York (1969)
Byrd, R.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Physicists. Springer, Berlin (1954)
Coleman, B.D., Olson, W.K., Swigon, D.: Theory of sequence-dependent DNA elasticity. J. Chem. Phys. 118, 7127–7140 (2003)
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., Tobias, I., Swigon, D.: Theory of the influence of end conditions on self-contact in DNA loops. J. Chem. Phys. 103, 9101–9109 (1995)
Fain, B., Rudnick, J., Östlund, S.: Conformations of linear DNA. Phys. Rev. E 55, 7364–7368 (1997)
Fain, B., Rudnick, J.: Conformations of closed DNA. Phys. Rev. E 60, 7239–7252 (1999)
Hagerman, P.J.: Flexibility of DNA. Ann. Rev. Biophys. Biophys. Chem. 17, 265–286 (1988)
Hao, M.-H., Olson, W.K.: Global equilibrium configurations of supercoiled DNA. Macromolecules 22, 3292–3303 (1989)
Hu, K.: A differential-geometric interpretation of Kirchhoff’s elastic rods. J. Math. Phys. 40, 3341–3352 (1999)
Hu, K.: Writhe of DNA induced by a terminal twist. Bull. Math. Biol. 67, 197–209 (2005) (with Erratum ibid. 67, 1155 (2005))
Le Bret, M.: Catastrophic variation of twist and writhing of circular DNAs with constraint. Biopolymers 18, 1709–1725 (1979)
Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, 4th edn. Dover, New York (1944)
Manning, R.S., Maddocks, J.H., Kahn, J.D.: A continuum rod model of sequence-dependent DNA structure. J. Chem. Phys. 105, 5626–5646 (1996)
Marini, J.C., Levene, S.D., Crothers, D.M., Englund, P.T.: A bent helix in kinetoplast DNA. Cold Spring Harbor Symp. Quant. Biol. 47, 279–283 (1982)
Olson, W.K.: Simulating DNA at low resolution. Curr. Opin. Struck. Biol. 6, 242–256 (1996)
Qian, H., White, J.H.: Terminal twist induced continuous writhe of a circular rod with intrinsic curvature. J. Biomol. Struct. Dyn. 16, 663–669 (1998)
Qian, H., White, J.H.: Twist induced abrupt writhe of naturally straight arch with induced curvature (unpublished)
Shi, Y., Hearst, J.E.: The Kirchhoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling. J. Chem. Phys. 101, 5184–5200 (1994)
Swigon, D., Coleman, B.D., Tobias, I.: The elastic rod model for DNA and its application to the tertiary structure of DNA minicircles in mononucleosomes. Biophys. J. 74, 2515–2530 (1998)
Tobias, I., Coleman, B.D., Lembo, M.: A class of exact dynamic solutions in the elastic rod model of DNA with implications for the theory of fluctuations in the torsional motion of plasmids. J. Chem. Phys. 105, 2517–2526 (1996)
Tobias, I., Coleman, B.D., Olson, W.K.: The dependence of DNA tertiary structure on end conditions: Theory and implications for topological transitions. J. Chem. Phys. 101, 10990–10996 (1994)
Tobias, I., Olson, W.K.: The effect of intrinsic curvature on supercoiling: predictions of elasticity theory. Biopolymers 33, 639–646 (1993)
Ulanovsky, L., Bodner, M., Trifonov, E.N., Choder, M.: Curved DNA: Design, synthesis, and circularization. Proc. Natl. Acad. Sci. USA 83, 862–866 (1986)
Wadati, M., Tsuru, H.: Elastic model of looped DNA. Physica D 21, 213–226 (1986)
Watson, J.D., Crick, F.H.C.: Molecular structure of nucleic acids. A structure for deoxyribose nucleic acid. Nature 171, 737–738 (1953)
Westcott, T.P., Tobias, I., Olson, W.K.: Elasticity theory and numerical analysis of DNA supercoiling: An application to DNA looping. J. Phys. Chem. 99, 17926–17935 (1995)
White, J.H., Bauer, W.R.: Finite-element analysis of the displacement of closed DNA loops under torsional stress. Philos. Trans. R. Soc. A 362, 1335–1353 (2004)
White, J.H., Lund, R.A., Bauer, W.R.: Twist, writhe, and geometry of a DNA loop containing equally spaced coplanar bends. Biopolymers 38, 235–250 (1996)
White, J.H., Lund, R.A., Bauer, W.R.: Effect of salt-dependent stiffness on the conformation of a stressed DNA loop containing initially coplanar bends. Biopolymers 49, 605–619 (1999)
Wu, H.-M., Crothers, D.M.: The locus of sequence-directed and protein-induced DNA bending. Nature 83, 509–513 (1984)
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Cheng, LT., Hu, K. On Solutions to Euler-Lagrange Equations Governing Isotropic, Homogeneous, Naturally Curved Kirchhoff’s Elastic Rods. J Elast 102, 1–14 (2011). https://doi.org/10.1007/s10659-010-9257-6
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DOI: https://doi.org/10.1007/s10659-010-9257-6
Keywords
- Naturally curved rods
- Total energy
- Twisting density
- Elastic rod model for DNA
- DNA with intrinsic curvature