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2017 | OriginalPaper | Buchkapitel

3. Link, Writhe, and Twist

verfasst von : Oliver M. O’Reilly

Erschienen in: Modeling Nonlinear Problems in the Mechanics of Strings and Rods

Verlag: Springer International Publishing

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Abstract

We review the concepts of writhe, twist, and linking as applied to space curves and ribbons. The application of these concepts to DNA is also discussed.

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Vorheriges Kapitel Applications of the Mechanics of a String

Nächstes Kapitel Theory of the Elastica and a Selection of Its Applications

We refer the reader to Bergou et al. [22] and Hanson [155, Chapter 20] for details on the numerical computation of the Bishop frame for space curves.

The total surface area of a unit sphere is 4π. In topology, the area A in Figure 3.2 is known as a solid angle.

Our historical comments in this section are based entirely on the (recent) insightful papers by Epple [95, 96] and Ricca and Nipoti [301]. The latter paper contains a translation of the page in Gauss’ notebook where Eqn. (3.28) is presented as well as copies of letters from Maxwell to Tait discussing the linking number.

Our convention for writing \(L_{\mbox{ k}}\left (\mathcal{S}_{1},\mathcal{S}_{2}\right )\) is taken from Spivak [329, Problem 8.28, Page 402] and differs from Gauss’ original prescription by a minus sign. As a result, our computations using Eqn. (3.28), such as the results shown in Figure 3.9, agree with those found by counting the signed crossings using Eqn. (3.34).

These methods are discussed in several places in the literature and summarized in Rolfsen’s text [303, Chapter 5, Section D].

Fuller’s version of Eqn. (3.38) differs from ours in that \(T_{\text{w}}\left (\mathcal{S},\mathbf{e}_{n}\right )\) is replaced by the more general case \(T_{\text{w}}\left (\mathcal{S},\mathbf{u}\right )\) in Eqn. (3.38). Alternative proofs of Fuller [111, Eqn. (6.3)] can be found in Aldinger et al. [7] and Kamien [176].

The parameters for this curve are discussed in Section 1.3.4.

The orientability condition on the ribbon is satisfied when u(s) = u(s + ℓ) where s ∈ [0, ℓ] on \(\mathcal{S}\). Thus, the ribbon is not a Möbius strip.

As emphasized in [7], the domain of integration excludes those points s ₁ = s ₂ where the integrand becomes unbounded.

The primary reference for our summary of the methods used to compute the writhing number is [7]. We also recommend the later works [20, 21, 85, 169, 176] for helpful perspectives and insights on this topic.

The curves considered in [360] are discussed in Exercise 3.9.

See [169, 360] and references therein.

We refer the reader to [239] for a discussion of Călugăreanu’s legacy and the roles played by Călugăreanu’s theorem and Eqn. (3.60).

For examples, see [18, 360, 361].

Underwound is also termed negatively supercoiled in contrast to the case σ > 0 which is termed positively supercoiled.

This result was first established by Maxwell [95, 301].

The curve \(\mathcal{S}_{2}\) is not identical to the curve \(\mathcal{S}_{2}\) that is defined in Eqn. (3.52)₂.

Agoston, M.K.: Computer Graphics and Geometric Modelling: Mathematics. Springer-Verlag, London (2005). URL http://dx.doi.org/10.1007/b138899

Aldinger, J., Klapper, I., Tabor, M.: Formulae for the calculation and estimation of writhe. Journal of Knot Theory and its Ramifications 4 (3), 343–372 (1995). URL http://dx.doi.org/10.1142/S021821659500017X

18.

Bauer, W.R., Lund, R.A., White, J.H.: Twist and writhe of a DNA loop containing intrinsic bends. Proceedings of the National Academy of Sciences 90 (3), 833–837 (1993)CrossRef

20.

Berger, M.A.: Topological quantities: Calculating winding, writhing, linking, and higher order invariants. In: R.L. Ricca (ed.) Lectures on Topological Fluid Mechanics, Lecture Notes in Mathematics, pp. 75–97. Springer-Verlag, Berlin, Heidelberg (2009). URL http://dx.doi.org/10.1007/978-3-642-00837-5_2

21.

Berger, M.A., Prior, C.: The writhe of open and closed curves. Journal of Physics A: Mathematical and General 39 (26), 8321 (2006). URL http://dx.doi.org/10.1088/0305-4470/39/26/005

22.

Bergou, M., Wardetzky, M., Robinson, S., Audoly, B., Grinspun, E.: Discrete elastic rods. ACM Transactions on Graphics (SIGGRAPH) 27 (3), 63:1–63:12 (2008)

28.

Bishop, R.L.: There is more than one way to frame a curve. The American Mathematical Monthly 82 (3), 246–251 (1975). URL http://dx.doi.org/10.2307/2319846

40.

Bustamante, C., Bryant, Z., Smith, S.B.: Ten years of tension: Single-molecule DNA mechanics. Nature 421, 423–427 (2003). URL http://dx.doi.org/10.1038/nature01405

42.

Călugăreanu, G.: L’intégrale de Gauss et l’analyse des nœuds tridimensionnels. Revue Roumaine Mathématique Pures et Appliqué 4, 5–20 (1959)

43.

Călugăreanu, G.: Sur les classes d’isotopie des nœuds tridimensionnels et leurs invariants. Czechoslovak Mathematical Journal 11 (86), 588–625 (1961)MathSciNetMATH

44.

Călugăreanu, G.: Sur les enlacements tridimensionnels des courbes fermées. Comptes Rendus Academie R. P. Romîne 11, 829–832 (1961)MATH

67.

Coleman, B.D., Swigon, D.: Theory of supercoiled elastic rings with self-contact and its application to DNA plasmids. Journal of Elasticity 60 (3), 173–221 (2000). URL http://dx.doi.org/10.1023/A:1010911113919

68.

Coleman, B.D., Swigon, D.: Theory of self-contact in Kirchhoff rods with applications to supercoiling of knotted and unknotted DNA plasmids. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 362 (1820), 1281–1299 (2004). URL http://dx.doi.org/10.1098/rsta.2004.1393

70.

Coleman, B.D., Swigon, D., Tobias, I.: Elastic stability of DNA configurations. II. Supercoiled plasmids with self-contact. Physical Review E 61, 759–770 (2000). URL http://dx.doi.org/10.1103/PhysRevE.61.759

77.

Crick, F.H.: Linking numbers and nucleosomes. Proceedings of the National Academy of Sciences 73 (8), 2639–2643 (1976). URL http://www.pnas.org/content/73/8/2639.abstract

83.

Daune, M.: Molecular Biophysics: Structures in Motion. Oxford University Press, New York (1999). Translated from the French by W. J. Duffin.

85.

Dennis, M.R., Hannay, J.H.: Geometry of Călugăreanu’s theorem. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 461 (2062), 3245–3254 (2005). URL http://dx.doi.org/10.1098/rspa.2005.1527

95.

Epple, M.: Orbits of asteroids, a braid, and the first link invariant. The Mathematical Intelligencer 20 (1), 45–52 (1998). URL http://dx.doi.org/10.1007/BF03024400

96.

Epple, M.: Topology, matter, and space, I: Topological notions in 19th-century natural philosophy. Archive for History of Exact Sciences 52 (4), 297–392 (1998). URL http://www.jstor.org/stable/41134050

110.

Fuller, F.B.: The writhing number of a space curve. Proceedings of the National Academy of Sciences 68, 815–819 (1971). URL http://www.pnas.org/content/68/4/815.abstract

111.

Fuller, F.B.: Decomposition of the linking number of a closed ribbon: A problem from molecular biology. Proceedings of the National Academy of Sciences 75 (8), 3557–3561 (1978). URL http://dx.doi.org/10.1073/pnas.75.8.3557

124.

Goyal, S., Perkins, N.C.: Looping mechanics of rods and DNA with non-homogeneous and discontinuous stiffness. International Journal of Non-Linear Mechanics 43 (10), 1121–1129 (2008). URL http://dx.doi.org/10.1016/j.ijnonlinmec.2008.06.013

144.

Guggenheimer, H.: Computing frames along a trajectory. Computer Aided Geometric Design 6 (1), 77–78 (1989). URL http://dx.doi.org/10.1016/0167-8396(89)90008-3

155.

Hanson, A.J.: Visualizing Quaternions. Morgan Kaufmann, San Francisco, CA; Amsterdam; Boston (2006)

169.

Hoffman, K.A., Manning, R.S., Maddocks, J.H.: Link, twist, energy, and the stability of DNA minicircles. Biopolymers 70 (2), 145–157 (2003). URL http://dx.doi.org/10.1002/bip.10430

176.

Kamien, R.D.: Local writhing dynamics. The European Physical Journal B 1 (1), 1–4 (1998). URL http://dx.doi.org/10.1007/s100510050145

188.

Kreyszig, E.: Differential Geometry, Revised and reprinted edn. Toronto University Press, Toronto (1964)MATH

196.

Langer, J., Singer, D.A.: Lagrangian aspects of the Kirchhoff elastic rod. SIAM Review 38 (4), 605–618 (1996). URL http://dx.doi.org/10.1137/S0036144593253290

210.

Livingston, C.: Knot Theory. Mathematical Association of America, Washington, DC (1993)MATH

213.

Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, fourth edn. Cambridge University Press, Cambridge (1927)

233.

Maxwell, J.C.: A Treatise on Electricity and Magnetism, vol. 2, third edn. Clarendon Press, Oxford (1892)

234.

McConnell, A.J.: Applications of the Absolute Differential Calculus. Blackie and Son, London (1947). Corrected reprinted edition

235.

McMillen, T., Goriely, A.: Tendril perversion in intrinsically curved rods. Journal of Nonlinear Science 12 (3), 241–281 (2002). URL http://dx.doi.org/10.1007/s00332-002-0493-1

239.

Moffatt, H.K., Ricca, R.L.: Helicity and the Călugăreanu invariant. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 439 (1906), 411–429 (1992). URL http://www.jstor.org/stable/52228

241.

Murasugi, K.: Knot theory and its applications. Birkhäuser Boston Inc., Boston, MA (1996). Translated from the 1993 Japanese original by Bohdan KurpitaMATH

248.

Neukirch, S., Starostin, E.L.: Writhe formulas and antipodal points in plectonemic DNA configurations. Physical Review E 78 (4), 041,912, 9 (2008). URL http://dx.doi.org/10.1103/PhysRevE.78.041912

257.

Oprea, J.: Differential Geometry and its Applications, second edn. Pearson Prentice Hall, Upper Saddle River, NJ (2003)

293.

Pohl, W.F.: The self-linking number of a closed space curve. Journal of Mathematics and Mechanics 17, 975–985 (1967/1968)

294.

Pohl, W.F.: DNA and differential geometry. The Mathematical Intelligencer 3 (1), 20–27 (1980/81). URL http://dx.doi.org/10.1007/BF03023391

301.

Ricca, R.L., Nipoti, B.: Gauss’ linking number revisited. Journal of Knot Theory and its Ramifications 20 (10), 1325–1343 (2011). URL http://dx.doi.org/10.1142/S0218216511009261

303.

Rolfsen, D.: Knots and Links, Mathematics Lecture Series, vol. 7. Publish or Perish, Inc., Houston, Texas (1990). Corrected reprint of the 1976 original

314.

Sarkar, A., Léger, J.F., Chatenay, D., Marko, J.F.: Structural transitions in DNA driven by external force and torque. Physical Review E 63 (5), 051,903 (2001). URL http://link.aps.org/doi/10.1103/PhysRevE.63.051903

329.

Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. I, second edn. Publish or Perish, Inc., Wilmington, Delaware (1979)

335.

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. Biophysical Journal 74 (5), 2515–2530 (1998). URL http://dx.doi.org/10.1016/S0006-3495(98)77960-3

337.

Tait, P.G.: Note on the measure of beknottedness. Proceedings of the Royal Society of Edinburgh 9, 289–298 (1877–1878). URL http://www.maths.ed.ac.uk/~aar/papers/beknot.pdf

341.

Thomson, W., Tait, P.G.: Treatise on Natural Philosophy. Oxford University Press, Oxford (1867)MATH

359.

White, J.H.: Self-linking and the Gauss integral in higher dimensions. American Journal of Mathematics 91, 693–728 (1969). URL http://www.jstor.org/stable/2373348

360.

White, J.H., Bauer, W.R.: Calculation of the twist and the writhe for representative models of DNA. Journal of Molecular Biology 189 (2), 329–341 (1986). URL http://dx.doi.org/10.1016/0022-2836(86)90513-9

361.

White, J.H., Bauer, W.R.: Applications of the twist difference to DNA structural analysis. Proceedings of the National Academy of Sciences 85 (3), 772–776 (1988). URL http://www.pnas.org/content/85/3/772.full.pdf

Titel: Link, Writhe, and Twist
verfasst von: Oliver M. O’Reilly
Verlag: Springer International Publishing
Buch: Modeling Nonlinear Problems in the Mechanics of Strings and Rods
Print ISBN: 978-3-319-50596-1

Electronic ISBN: 978-3-319-50598-5

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-50598-5_3

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