Abstract
Supercoiling of a closed circular DNA rod may result from an application of terminal twist to the DNA rod by cutting the rod, rotating one of the cut faces as the other being fixed and then sealing the cut. According to White's formula, DNA supercoiling is probably accompanied by a writhe of the DNA axis. Deduced from the elastic rod model for DNA structure, an intrinsically straight closed circular DNA rod does not writhe as subject to a terminal twist, until the number of rotation exceeds a rod-dependent threshold. By contrast, a closed circular DNA rod with intrinsic curvature writhes instantly as subject to a terminal twist. This noteworthy character in fact belongs to many intrinsically curved DNA rods. By solving the dynamic equations, the linearization of the Euler–Lagrange equations governing intrinsically curved DNA rods, this paper shows that almost every clamped-end intrinsically curved DNA rod writhes instantly when subject to a terminal twist (clamped-end DNA rods include closed circular DNA rods and topological domains of open DNA rods). In terms of physical quantities, the exceptions are identified with points in ℝ6 whose projections onto ℝ5 (through ignoring the total energy density of a rod) form a subset of a quadratic hypersurface. This paper also suggests that the terminal twist induced writhe is due to the elasticity and the clamped-end boundary conditions of the DNA rods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bates, A.D., Maxwell, A., 2005. DNA Topology, 2nd ed. Oxford University Press, Oxford.
Bauer, W.R., Lund, R.A., White, J.H., 1993. Twist and writhe of a DNA loop containing intrinsic bends. Proc. Natl. Acad. Sci. U.S.A. 90, 833–837.
Benham, C.J., 1989. Onset of writhing in circular elastic polymers. Phys. Rev. A 39, 2582–2586.
Birkhoff, G., Rota, G.-C., 1969. Ordinary Differential Equations, 2nd ed. Wiley New York.
Gasser, S.M., Laemmli, U.K., 1987. A glimpse at chromosoma order. Trends Genet. 3, 16–21.
Geanacopoulos, M., Vasmatzis, G., Zhurkin, V.B., Adhya, S., 2001. Gal repressosome contains an antiparallel DNA loop. Nature Struct. Biol. 8, 432–436.
Hu, K., 1999. A differential-geometric interpretation of Kirchhoff’s elastic rods. J. Math. Phys. 40, 3341–3352.
Hu, K., 2003. Buckling of some isotropic, intrinsically curved elasticas induced by a terminal twist. Appl. Math. Lett. 16, 193–197.
Krämer, H., Niemöller, M., Amouyal, M., Revêt, B., von Wilcken-Bergmann, Müller-Hill, B., 1987. Lac repressor forms loops with linear DNA carrying two suitably spaced lac operators. EMBO J. 6, 1481–1491.
Krämer, H., Amouyal, M., Nordheim, A., Müller-Hill, 1988. DNA supercoiling changes the spacing requirement of two lac operators for DNA loop formation with Lac repressor. EMBO J. 7, 547–556.
Kramer, P.R., Sinden, R.R., 1997. Measurement of unrestrained negative supercoiling and topological domain size in living human cells. Biochemistry 36, 3151–3158.
Le Bret, M., 1979. Catastrophic variation of twist and writhing of circular DNAs with constraint. Biopolymers 18, 1709–1725.
Lobell, R.B., Schleif, R.F., 1990. DNA looping and unlooping by AraC protein. Science 250, 528–532.
Love, A.E.H., 1944. A Treatise on the Mathematical Theory of Elasticity. 4th ed. Dover, New York.
Marini, J.C., Levene, S.D., Crothers, D.M., Englund, P.T., 1982. A bent helix in kinetoplast DNA. Cold Spring Harbor Symp. Quant. Biol. 47, 279–283.
Nurse, P., Levine, C., Hassing, H., Marians, K.J., 2003. Topoisomerase III can server as the cellular decatenase in Escherichia coli. J. Biol. Chem. 278, 8653–8660.
Pollock, T.J., Nash, H.A., 1983. Knotting of DNA caused by a genetic rearrangement. Evidence for a nucleosome-like structure in site-specific recombination of bacteriophage lambda. J. Mol. Biol. 170, 1–18.
Qian, H., White, J.H., 1998. Terminal twist induced continuous writhe of a circular rod with intrinsic curvature. J. Biomol. Struct. Dyn. 16, 663–669.
Schleif, R., 1992. DNA looping. Annu. Rev. Biochem. 61, 199–223.
Su, W., Porter, S., Kustu, S., Echols, H., 1990. DNA-looping and enhancer activity: Association between DNA-bound NtrC activator and RNA polymerase at the bacterial glnA promoter. Proc. Natl. Acad. Sci. U.S.A. 87, 5504–5508.
Tobias, I., Coleman, B.C., Lembo, M., 1996. 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.
Ulanovsky, L., Bodner, M., Trifonov, E.N., Choder, M., 1986. Curved DNA: Design, synthesis, circularization. Proc. Natl. Acad. Sci. U.S.A. 83, 862–866.
Westcott, T.P., Tobias, I., Olson, W.K., 1995. Elasticity theory and numerical analysis of DNA supercoiling: An application to DNA looping. J. Phys. Chem. 99, 17926–17935.
White, J.H. (1969). Self-linking and the Gauss integral in higher dimensions. Am. J. Math. 91, 693–728.
White, J.H., Lund, R.A., Bauer, W.R., 1996. Twist, writhe, and geometry of a DNA loop containing equally spaced coplanar bends. Biopolymers 38, 235–250.
Whitson, P.A., Hsieh, W.-T., Wells, R.D., Matthews, K.S., 1987. Influence of supercoiling and sequence context on operator DNA binding with lac repressor. J. Biol. Chem. 262, 14592–14599.
Wu, H.-M., Crothers, D.M., 1984. The locus of sequence-directed and protein-induced DNA bending. Nature 83, 509–513.
Zajac, E.E. (1962). Stability of two planar loop elasticas. J. Appl. Mech. (Trans. ASME Ser. E) 29, 136–142.
Author information
Authors and Affiliations
Corresponding author
Additional information
To my sister for her 50th birthday.
Rights and permissions
About this article
Cite this article
Hu, K. Terminal Twist-Induced Writhe of DNA with Intrinsic Curvature. Bull. Math. Biol. 69, 1019–1030 (2007). https://doi.org/10.1007/s11538-006-9156-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-006-9156-y