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The Mathematics of DNA Structure, Mechanics, and Dynamics

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Mathematics of DNA Structure, Function and Interactions

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 150))

Abstract

A brief review is given of the main concepts, ideas, and results in the fields of DNA topology, elasticity, mechanics and statistical mechanics. Discussion in- cludes the notions of the linking number, writhe, and twist of closed DNA, elastic rod models, sequence-dependent base-pair level models, statistical models such as helical worm-like chain and freely jointed chain, and dynamical simulation procedures. Experimental methods that lead to the development of the models and the implications of the models are also discussed. Emphasis is placed on illustrating the breadth of approaches and the latest developments in the field, rather than the depth and completeness of exposition.

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References

  1. J. Aldinger, I. Klapper, and M. Tabor, Formulae for the calculation and estimation of writhe, J. Knot Theory Ramifications, 4(1995), pp. 343–372.

    Article  Google Scholar 

  2. S.S. Antman, Nonlinear problems of elasticity, Vol. 107 of Applied Mathematical Sciences, Springer, New York, second ed., 2005.

    Google Scholar 

  3. S.S. Antman, and C.S. Kenney, Large buckled states of nonlinearly elastic rods under torsion, thrust, and gravity, Arch. Rational Mech. Anal., 76(1981), pp. 289–338.

    Article  Google Scholar 

  4. S.S. Antman and T.-P. Liu, Travelling waves in hyperelastic rods, Quart. Appl. Math., 36(1979), pp. 377–399.

    Google Scholar 

  5. J. Arsuaga, M. v´azquez, S. Trigueros, D. Sumners, and J. Roca, Knotting probability of DNA molecules confined in restricted volumes: DNA knotting in phage capsids, Proc. Natl. Acad. Sci. U.S.A., 99(2002), pp. 5373–5377.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  6. A. Balaeff, C.R. Koudella, L. Mahadevan, and K. Schulten, Modelling DNA loops using continuum and statistical mechanics, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 362(2004), pp. 1355–1371.

    Article  CAS  Google Scholar 

  7. A. Balaeff, L. Mahadevan, and K. Schulten, Modeling DNA loops using the theory of elasticity, Phys. Rev. E (3), 73(2006), pp. 031919, 23.

    Article  CAS  Google Scholar 

  8. A. Barbic and D.M. Crothers, Comparison of analyses of DNA curvature, J. Biomol. Struct. Dyn., 21(2003), pp. 89–97.

    Article  CAS  PubMed  Google Scholar 

  9. A.D. Bates and A. Maxwell, DNA topology, Oxford University Press, 1993.

    Google Scholar 

  10. C. Baumann, S. Smith, V. Bloomfield, and C. Bustamante, Ionic effects on the elasticity of single DNA molecules, Proc. Natl. Acad. Sci. U.S.A., 94(1997), pp. 6185–6190.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  11. C. Benham and S. Mielke, DNA mechanics, Annu Rev Biomed Eng, 7(1997), pp. 21–53.

    Google Scholar 

  12. C.J. Benham, Elastic model of supercoiling, Proc. Natl. Acad. Sci. U.S.A., 74(1977), pp. 2397–2401.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  13. C.J. Benham,An elastic model of the large-scale structure of duplex DNA, Biopolymers, 18(1979), pp. 609–623.

    Google Scholar 

  14. C.J. Benham,Theoretical analysis of heteropolymeric transitions in superhelical DNA molecules of specified sequence., J. Chem. Phys., 92(1990), pp. 6294–6305.

    Google Scholar 

  15. C.J. Benham,Energetics of the strand separation transition in superhelical DNA, J. Mol. Biol., 225(1992), pp. 835–847.

    Google Scholar 

  16. C.J. Benham,Duplex destabilization in superhelical DNA is predicted to occur at specific transcriptional regulatory regions., J. Mol. Biol., 255(1996), pp. 425–434.

    Google Scholar 

  17. D.L. Beveridge, G. Barreiro, K.S. Byun, D.A. Case, T.E. Cheatham, S.B. Dixit, E. Giudice, F. Lankas, R. Lavery, J.H. Maddocks, R. Osman, E. Seibert, H. Sklenar, G. Stoll, K.M. Thayer, P. Varnai, and M.A. Young, Molecular dynamics simulations of the 136 unique tetranucleotide sequences of DNA oligonucleotides. I. Research design and results on d(CpG) steps, Biophys. J., 87(2004), pp. 3799–3813.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  18. Y.Y. Biton, B.D. Coleman, and D. Swigon, On bifurcations of equilibria of intrinsically curved, electrically charged, rod-like structures that model DNA molecules in solution, J. Elasticity, 87(2007), pp. 187–210.

    Article  Google Scholar 

  19. V.A. Bloomfield and I. Rouzina, Use of Poisson-Boltzmann equation to analyze ion binding to DNA, Meth. Enzymol., 295(1998), pp. 364–378.

    Article  CAS  Google Scholar 

  20. A. Bolshoy, P. Mcnamara, R.E. Harrington, AND E.N. Trifonov, Curved DNA without A-A: experimental estimation of all 16 DNA wedge angles, Proc. Natl. Acad. Sci. U.S.A., 88(1991), pp. 2312–2316.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  21. A.H. Boschitsch and M.O. Fenley, Hybrid boundary element and finite difference method for solving the nonlinear Poisson-Boltzmann equation, J. Comput. Chem., 25(2004), pp. 935–955.

    Article  CAS  PubMed  Google Scholar 

  22. C. Bouchiat and M. Mezard, Elasticity model of a supercoiled DNA molecule, Phys. Rev. Lett., 80(1998), pp. 1556–1559.

    Article  CAS  Google Scholar 

  23. C. Bouchiat, M. Wang, J. Allemand, T. Strick, S. Block, and V. Croquette, Estimating the persistence length of a worm-like chain molecule from force-extension measurements, Biophys. J., 76(1999), pp. 409–413.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  24. Z. Bryant, M.D. Stone, J. Gore, S.B. Smith, N.R. Cozzarelli, and C. Bustamante, Structural transitions and elasticity from torque measurements on DNA, Nature, 424(2003), pp. 338–341.

    Article  CAS  PubMed  Google Scholar 

  25. G.R. Buck and E.L. Zechiedrich, DNA disentangling by type-2 topoisomerases, J. Mol. Biol., 340(2004), pp. 933–939.

    Article  CAS  PubMed  Google Scholar 

  26. C. Bustamante, Z. Bryant, and S.B. Smith, Ten years of tension: singlemolecule DNA mechanics, Nature, 421(2003), pp. 423–427.

    Article  PubMed  CAS  Google Scholar 

  27. C. Bustamante, S.B. Smith, J. Liphardt, and D. Smith, Single-molecule studies of DNA mechanics, Curr. Opin. Struct. Biol., 10(2000), pp. 279–285.

    Article  CAS  PubMed  Google Scholar 

  28. C.R. Calladine and H.R. Drew, Understanding DNA, Academic Press, 1992.

    Google Scholar 

  29. G. CĂalugĂareanu, Sur les classes d’isotopie des noeuds tridimensionnels et leurs invariants, Czechoslovak Math. J., 11(86) (1961), pp. 588–625.

    Article  Google Scholar 

  30. G. Charvin, J.F. Allemand, T.R. Strick, D. Bensimon, and V. Croquette, Twisting DNA: single molecule studies, Cont. Phys., 45(2004), pp. 383–403.

    Article  CAS  Google Scholar 

  31. G. Chirico and J. Langowski, Calculating hydrodynamic properties of DNA through a second-order Brownian dynamics algorithm, Macromolecules, 25(1992), pp. 769–775.

    Article  CAS  Google Scholar 

  32. G. Chirico and J. Langowski, Kinetics of DNA supercoiling studied by Brownian dynamics simulation, Biopolymers, 34(1994), pp. 211–225.

    Google Scholar 

  33. G. Chirico and J. Langowski, Brownian dynamics simulations of supercoiled DNA with bent sequences, Biophys. J., 71(1996), pp. 955–971.

    Google Scholar 

  34. G.S. Chirikjian and Y. Wang, Conformational statistics of stiff macromolecules as solutions to partial differential equations on the rotation and motion groups, Phys. Rev. E Stat. Phys. Plasmas. Fluids Relat. Interdiscip. Top- ics, 62(2000), pp. 880–892.

    CAS  Google Scholar 

  35. T. Cloutier and J. Widom, Spontaneous sharp bending of double-stranded DNA, Mol. Cell, 14(2004), pp. 355–362.

    Article  CAS  PubMed  Google Scholar 

  36. T. Cloutier and J. Widom, DNA twisting flexibility and the formation of sharply looped protein-DNA complexes, Proc. Natl. Acad. Sci. U.S.A., 102(2005), pp. 3645–3650.

    Google Scholar 

  37. B.D. Coleman, E.H. Dill, M. Lembo, Z. Lu, and I. Tobias, On the dynamics of rods in the theory of Kirchhoff and Clebsch, Arch. Rational Mech. Anal., 121(1992), pp. 339–359.

    Article  Google Scholar 

  38. B.D. Coleman, W.K. Olson, and D. Swigon, Theory of sequence-dependent DNA elasticity, J. Chem. Phys., 118(2003), pp. 7127–7140.

    Article  CAS  Google Scholar 

  39. B.D. Coleman and D. Swigon, Theory of supercoiled elastic rings with selfcontact and its application to DNA plasmids, J. Elasticity, 60 (2000), pp. 173–221 (2001).

    Article  Google Scholar 

  40. B.D. Coleman and D. Swigon, Theory of self-contact in Kirchhoff rods with applications to supercoiling of knotted and unknotted DNA plasmids, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 362(2004), pp. 1281–1299.

    Google Scholar 

  41. B.D. Coleman, D. Swigon, and I. Tobias, Elastic stability of DNA configurations. II. Supercoiled plasmids with self-contact, Phys. Rev. E (3), 61(2000), pp. 759–770.

    Article  CAS  Google Scholar 

  42. R. Courant, Differential and Integral Calculus, Vol. 2, Blackie, London, 1936.

    Google Scholar 

  43. D.M. Crothers, J. Drak, J.D. Kahn, and S.D. Levene, DNA bending, flexibility, and helical repeat by cyclization kinetics, Meth. Enzymol., 212(1992), pp. 3–29.

    Article  CAS  Google Scholar 

  44. L. Czapla, D. Swigon, and W.K. Olson, Sequence-dependent effects in the cyclization of short DNA, J. Chem. Theory Comput., 2(2006), pp. 685–695.

    Article  CAS  PubMed  Google Scholar 

  45. L. Czapla, D. Swigon, and W.K. Olson, Effects of the nucleoid protein HU on the structure, flexibility, and ringclosure properties of DNA deduced from monte-carlo simulations, submitted to J. Mol. Biol. (2008).

    Google Scholar 

  46. I.K. Darcy and D.W. Sumners, A strand passage metric for topoisomerase action, in KNOTS ’96 (Tokyo), World Sci. Publ., River Edge, NJ, 1997, pp. 267–278.

    Google Scholar 

  47. I.K. Darcy and D.W. Sumners, Rational tangle distances on knots and links, Math. Proc. Cambridge Philos. Soc., 128(2000), pp. 497–510.

    Google Scholar 

  48. Y. Diao, J.C. Nardo, and Y. Sun, Global knotting in equilateral random polygons, J. Knot Theory Ram., 10(2001), pp. 597–607.

    Article  Google Scholar 

  49. D.J. Dichmann, Y. Li, and J.H. Maddocks, Hamiltonian formulations and symmetries in rod mechanics, in Mathematical approaches to biomolecular structure and dynamics (Minneapolis, MN, 1994), Vol. 82 of IMA Vol. Math. Appl., Springer, New York, 1996, pp. 71–113.

    Book  Google Scholar 

  50. D.J. Dichmann, J.H. Maddocks, and R.L. Pego, Hamiltonian dynamics of an elastica and the stability of solitary waves, Arch. Rational Mech. Anal., 135(1996), pp. 357–396.

    Article  Google Scholar 

  51. [52] E.H. Dill, Kirchhoff’s theory of rods, Arch. Hist. Exact Sci., 44(1992), pp. 1–23.

    Article  Google Scholar 

  52. [52] S.B. Dixit, D.L. Beveridge, D.A. Case, T.E. Cheatham, E. Giudice, F. Lankas, R. Lavery, J.H. Maddocks, R. Osman, H. Sklenar, K.M. Thayer, and P. Varnai, Molecular dynamics simulations of the 136 unique tetranucleotide sequences of DNA oligonucleotides. II: sequence context effects on the dynamical structures of the 10 unique dinucleotide steps, Bio- phys. J., 89(2005), pp. 3721–3740.

    CAS  Google Scholar 

  53. L.D. Doi and S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, 1988.

    Google Scholar 

  54. Q. Du, A. Kotlyar, and A. Vologodskii, Kinking the double helix by bending deformation, Nucleic Acids Res., 36(2008), pp. 1120–1128.

    Article  CAS  PubMed  Google Scholar 

  55. C. Ernst and D. W. Sumners, A calculus for rational tangles: applications to DNA recombination, Math. Proc. Cambridge Philos. Soc., 108(1990), pp. 489–515.

    Article  Google Scholar 

  56. M.D. Frank-kamenetskii, A.V. Lukashin, and A.V. Vologodskii, Statistical mechanics and topology of polymer chains, Nature, 258(1975), pp. 398–402.

    Article  CAS  PubMed  Google Scholar 

  57. F.B. Fuller, The writhing number of a space curve, Proc. Nat. Acad. Sci. U.S.A., 68(1971), pp. 815–819.

    Article  CAS  Google Scholar 

  58. P.B. Furrer, R.S. Manning, and J.H. Maddocks, DNA rings with multiple energy minima, Biophys. J., 79(2000), pp. 116–136.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  59. S. Gavryushov and P. Zielenkiewicz, Electrostatic potential of B-DNA: effect of interionic correlations, Biophys. J., 75(1998), pp. 2732–2742.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  60. J.A. Gebe, S.A. Allison, J.B. Clendenning, and J.M. Schurr, Monte Carlo simulations of supercoiling free energies for unknotted and trefoil knotted DNAs, Biophys. J., 68(1995), pp. 619–633.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  61. O. Gonzales and J.H. Maddocks, Extracting parameters for base-pair level models of DNA from molecular dynamics simulations, Theor. Chem. Acc., 106(2001), pp. 76–82.

    Article  Google Scholar 

  62. O. Gonzalez, J.H. Maddocks, F. Schuricht, and H. Von der mosel, Global curvature and self-contact of nonlinearly elastic curves and rods, Calc. Var. Partial Differential Equations, 14(2002), pp. 29–68.

    Article  Google Scholar 

  63. A. Goriely and M. Tabor, Nonlinear dynamics of filaments. IV. Spontaneous looping of twisted elastic rods, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454(1998), pp. 3183–3202.

    Article  Google Scholar 

  64. A. Goriely and M. Tabor, The nonlinear dynamics of filaments, Nonlinear Dynam., 21(2000), pp. 101–133. 316 DAVID SWIGON

    Google Scholar 

  65. S. Goyal, N.C. Perkins, and C.L. Lee, Nonlinear dynamics and loop formation in Kirchhoff rods with implications to the mechanics of DNA and cables, J. Comput. Phys., 209(2005), pp. 371–389.

    Article  CAS  Google Scholar 

  66. P.J. Hagerman, Flexibility of DNA, Annu Rev Biophys Biophys Chem, 17(1988), pp. 265–286.

    Article  CAS  PubMed  Google Scholar 

  67. K.A. Hoffman, Methods for determining stability in continuum elastic-rod models of DNA, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 362(2004), pp. 1301–1315.

    Article  Google Scholar 

  68. K.A. Hoffman, R.S. Manning, and J.H. Maddocks, Link, twist, energy, and the stability of DNA minicircles, Biopolymers, 70(2003), pp. 145–157.

    Article  CAS  PubMed  Google Scholar 

  69. D.S. Horowitz and J.C. Wang, Torsional rigidity of DNA and length dependence of the free energy of DNA supercoiling, J. Mol. Biol., 173(1984), pp. 75–91.

    Article  CAS  PubMed  Google Scholar 

  70. X. Hua, D. Nguyen, B. Raghavan, J. Arsuaga, and M. Vazquez, Random state transitions of knots: A first step towards modeling unknotting by type II topoisomerases, Topology Appl., 154(2007), pp. 1381–1397.

    Article  Google Scholar 

  71. J. Huang, T. Schlick, and A. Vologodskii, Dynamics of site juxtaposition in supercoiled DNA, Proc. Natl. Acad. Sci. U.S.A., 98(2001), pp. 968–973.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  72. N. Hud and J. Plavec, A unified model for the origin of DNA sequence-directed curvature, Biopolymers, 69(2003), pp. 144–158.

    Article  CAS  PubMed  Google Scholar 

  73. H. Jian, T. Schlick, and A. Vologodskii, Internal motion of supercoiled DNA: Brownian dynamics simulations of site juxtaposition, J. Mol. Biol., 284(1998), pp. 287–296.

    Article  CAS  PubMed  Google Scholar 

  74. F. Julicher, Supercoiling transitions of closed DNA, Phys. Rev. E, 49(1994), pp. 2429–2435.

    Article  CAS  Google Scholar 

  75. G. Kirchhoff, Über das Gleichgewicht und die Bewegung eines unendlich d¨unen elastischen Stabes, J. Reine angew. Math. (Crelle), 56(1859), pp. 285–313.

    Google Scholar 

  76. J.G. Kirkwood, On the theory of strong electrolyte solutions, J. Chem. Phys., 2(1934), pp. 767–781.

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  78. K. Klenin, H. Merlitz, and J. Langowski, A Brownian dynamics program for the simulation of linear and circular DNA and other wormlike chain polyelectrolytes, Biophys. J., 74(1998), pp. 780–788.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  79. K.V. Klenin, A.V. Vologodskii, V.V. Anshelevich, A.M. Dykhne, and M.D. FRANK-KAMENETSKII, Computer simulation of DNA supercoiling, J. Mol. Biol., 217(1991), pp. 413–419.

    Article  CAS  PubMed  Google Scholar 

  80. K.M. Kosikov, A.A. Gorin, X.J. Lu, W.K. Olson, and G.S. Manning, Bending of DNA by asymmetric charge neutralization: all-atom energy simulations, J. Am. Chem. Soc., 124(2002), pp. 4838–4847.

    Article  CAS  PubMed  Google Scholar 

  81. O. Kratky and G. Porod, Röontgenuntersuchung gelüster fadenmolek¨ule, Rec. Trav. Chim. Pays-Bas., 68(1949), pp. 1106–1123.

    Article  CAS  Google Scholar 

  82. L.D. Landau and E.M. Lifshitz, Theory of Elasticity, Pergamon Press, Oxford, 1959.

    Google Scholar 

  83. L.D. Landau and E.M. Lifshitz, Statistical Physics, Butterworth-Heinemann, 1984.

    Google Scholar 

  84. J. Langer and D.A. Singer, Knotted elastic curves in R 3, J. London Math. Soc. (2), 30(1984), pp. 512–520.

    Article  Google Scholar 

  85. F. Lankas, R. Lavery, and J. Maddocks, Kinking occurs during molecular dynamics simulations of small DNA minicircles, Structure, 14(2006), pp. 1527–1534.

    Article  CAS  PubMed  Google Scholar 

  86. C.L. Lawson, D. Swigon, K.S. Murakami, S.A. Darst, H.M. Berman, and R.H. Ebright, Catabolite activator protein: DNA binding and transcription activation, Curr. Opin. Struct. Biol., 14(2004), pp. 10–20.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  87. M. Le bret, Catastrophic variation of twist and writhing of circular DNAs with constraint?, Biopolymers, 18(1979), pp. 1709–1725.

    Article  PubMed  Google Scholar 

  88. M. Le bret, Twist and writhing in short circular DNAs according to first-order elasticity, Biopolymers, 23(1984), pp. 1835–1867.

    Google Scholar 

  89. S. Levene and D. Crothers, Topological distributions and the torsional rigidity of DNA. A Monte Carlo study of DNA circles, J. Mol. Biol., 189(1986), pp. 73–83.

    Article  CAS  PubMed  Google Scholar 

  90. S. D. Levene and D. M. Crothers, Ring closure probabilities for DNA fragments by Monte Carlo simulation, J. Mol. Biol., 189(1986), pp. 61–72.

    Article  CAS  PubMed  Google Scholar 

  91. S. Lim, A. Ferent, X.S. Wang, and C.S. Peskin, Dynamics of a closed rod with twist and bend in fluid, preprint.

    Google Scholar 

  92. A.E.H. Love, Treatise on the mathematical theory of elasticity, Cambridge Uni- versity Press, 1927.

    Google Scholar 

  93. J.H. Maddocks, Bifurcation theory, symmetry breaking and homogenization in continuum mechanics descriptions of DNA. Mathematical modelling of the physics of the double helix, in A celebration of mathematical modeling, Kluwer Acad. Publ., Dordrecht, 2004, pp. 113–136.

    Google Scholar 

  94. G.S. Manning, The molecular theory of polyelectrolyte solutions with applications to the electrostatic properties of polynucleotides, Q. Rev. Biophys., 11(1978), pp. 179–246.

    Article  CAS  PubMed  Google Scholar 

  95. R.S. Manning, J.H. Maddocks, and J.D. Kahn, A continuum rod model of sequence-dependent DNA structure, J. Chem. Phys., 105(1996), pp. 5626–5646.

    Article  CAS  Google Scholar 

  96. R.S. Manning, K.A. Rogers, and J.H. Maddocks, Isoperimetric conjugate points with application to the stability of DNA minicircles, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454(1998), pp. 3047–3074.

    Article  CAS  Google Scholar 

  97. J.F. Marko, DNA under high tension: Overstretching, undertwisting, and relaxation dynamics, Phys. Rev. E, 57(1998), pp. 2134–2149.

    Article  CAS  Google Scholar 

  98. J.F. Marko and E.D. Siggia, Fluctuations and supercoiling of DNA, Science, 265(1994), pp. 506–508.

    Article  CAS  PubMed  Google Scholar 

  99. J.F. Marko and E.D. Siggia, Stretching DNA, Macromolecules, 28(1995), pp. 8759–8770.

    Google Scholar 

  100. H. Merlitz, K. Rippe, K.V. Klenin, and J. Langowski, Looping dynamics of linear DNA molecules and the effect of DNA curvature: a study by Brownian dynamics simulation, Biophys. J., 74(1998), pp. 773–779.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  101. C. Micheletti, D. Marenduzzo, E. Orlandini, and D. Summers, Knotting of random ring polymers in confined spaces, J. Chem. Phys., 124(2006), p. 64903.

    Article  CAS  PubMed  Google Scholar 

  102. J. Moroz and P. Nelson, Torsional directed walks, entropic elasticity, and DNA twist stiffness, Proc. Natl. Acad. Sci. U.S.A., 94(1997), pp. 14418–14422.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  103. J.D. Moroz and P. Nelson, Torsional directed walks, entropic elasticity, and DNA twist stiffness, Proc. Natl. Acad. Sci. U.S.A., 94(1997), pp. 14418–14422.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  104. S. Neidle, Principles of nucleic acid structure, Elsevier, 2007.

    Google Scholar 

  105. S. Neukirch and M.E. Henderson, Classification of the spatial equilibria of the clamped elastica: symmetries and zoology of solutions, J. Elasticity, 68(2002), pp. 95–121 (2003). Dedicated to Piero Villaggio on the occasion of his 70th birthday.

    Article  Google Scholar 

  106. W.K. Olson, Simulating DNA at low resolution, Curr. Opin. Struct. Biol., 6(1996), pp. 242–256.

    Article  CAS  PubMed  Google Scholar 

  107. W.K. Olson, M. Bansal, S.K. Burley, R.E. Dickerson, M. Gerstein, S.C. Harvey, U. Heinemann, X.J. Lu, S. Neidle, Z. Shakked, H. Sklenar, M. Suzuki, C.S. Tung, E. Westhof, C. Wolberger, and H.M. Berman, A standard reference frame for the description of nucleic acid base-pair geometry, J. Mol. Biol., 313(2001), pp. 229–237.

    Article  CAS  PubMed  Google Scholar 

  108. W.K. Olson, A.A. Gorin, X.J. Lu, L.M. Hock, and V.B. Zhurkin, DNA sequence-dependent deformability deduced from protein-DNA crystal complexes, Proc. Natl. Acad. Sci. U.S.A., 95(1998), pp. 11163–11168.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  109. W.K. Olson, D. Swigon, and B.D. Coleman, Implications of the dependence of the elastic properties of DNA on nucleotide sequence, Philos Transact A Math Phys Eng Sci, 362(2004), pp. 1403–1422.

    Article  Google Scholar 

  110. M.J. Packer, M.P. Dauncey, and C.A. Hunter, Sequence-dependent DNA structure: tetranucleotide conformational maps, J. Mol. Biol., 295(2000), pp. 85–103.

    Article  CAS  PubMed  Google Scholar 

  111. L.J. Parkhurst, K.M. Parkhurst, R. Powell, J. Wu, and S. Williams, Time-resolved fluorescence resonance energy transfer studies of DNA bending in double-stranded oligonucleotides and in DNA-protein complexes, Biopolymers, 61(2001), pp. 180–200.

    Article  PubMed  CAS  Google Scholar 

  112. C. Rivetti, C. Walker, and C. Bustamante, Polymer chain statistics and conformational analysis of DNA molecules with bends or sections of different flexibility, J. Mol. Biol., 280(1998), pp. 41–59.

    Article  CAS  PubMed  Google Scholar 

  113. V. Rybenkov, N. Cozzarelli, and A. Vologodskii, Probability of DNA knotting and the effective diameter of the DNA double helix, Proc. Natl. Acad. Sci. U.S.A., 90(1993), pp. 5307–5311.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  114. V.V. Rybenkov, C. Ullsperger, A.V. Vologodskii, and N.R. Cozzarelli, Simplification of DNA topology below equilibrium values by type II topoisomerases, Science, 277(1997), pp. 690–693.

    Article  CAS  PubMed  Google Scholar 

  115. W. Saenger, Principles of Nucleic Acid Structure, Springer-Verlag, New York and Berlin, 1984.

    Book  Google Scholar 

  116. S. Sankararaman and J. Marko, Formation of loops in DNA under tension, Phys Rev E Stat Nonlin Soft Matter Phys, 71(2005), p. 021911.

    Article  PubMed  CAS  Google Scholar 

  117. T. Schlick, Modeling superhelical DNA: recent analytical and dynamic approaches, Curr. Opin. Struct. Biol., 5(1995), pp. 245–262.

    Article  CAS  PubMed  Google Scholar 

  118. T. Schlick, B. Li, and W.K. Olson, The influence of salt on DNA energetics and dynamics, Biophys. J., 67(1994), pp. 2146–2166.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  119. T. Schlick, W. Olson, T. Westcott, and J. Greenberg, On higher buckling transitions in supercoiled DNA, Biopolymers, 34(1994), pp. 565–597.

    Article  CAS  PubMed  Google Scholar 

  120. F. Schuricht, Global injectivity and topological constraints for spatial nonlinearly elastic rods, J. Nonlin. Science, 12(2002), pp. 423–444.

    Article  CAS  Google Scholar 

  121. J.M. Schurr, B.S. Fujimoto, P. Wu, and S.L., Fluorescence studies of nucleic acids: Dynamics, rigidities, and structures, in Biochemical Applications, Vol. 3 of Topics in Fluorescence Spectroscopy, Plenum Press, New York, 1992.

    Google Scholar 

  122. S. Shaw and J. Wang, Knotting of a DNA chain during ring closure, Science, 260(1993), pp. 533–536.

    Article  CAS  PubMed  Google Scholar 

  123. J. Shimada and H. Yamakawa, Ring-closure probabilities for twisted wormlike chains, Macromolecules, 17(1984), pp. 689–698.

    Article  CAS  Google Scholar 

  124. D. Shore and R. Baldwin, Energetics of DNA twisting. I. Relation between twist and cyclization probability, J. Mol. Biol., 170(1983), pp. 957–981.

    Article  CAS  PubMed  Google Scholar 

  125. D. Shore and R. Baldwin, Energetics of DNA twisting. II. Topoisomer analysis, J. Mol. Biol., 170(1983), pp. 983–1007.

    Google Scholar 

  126. D. Shore, J. LANG¨OWSKI, AND R. L. Baldwin, DNA flexibility studied by covalent closure of short fragments into circles, Proc. Natl. Acad. Sci. Usa, 78(1981), pp. 4833–4837.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  127. S. Smith, Y. Cui, and C. Bustamante, Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules, Science, 271(1996), pp. 795–799.

    Article  CAS  Google Scholar 

  128. S. Smith, L. Finzi, and C. Bustamante, Direct mechanical measurements of the elasticity of single DNA molecules by using magnetic beads, Science, 258(1992), pp. 1122–1126.

    Article  CAS  PubMed  Google Scholar 

  129. E.L. Starostin, Three-dimensional shapes of looped DNA, Meccanica, 31(1996), pp. 235–271.

    Article  Google Scholar 

  130. T. Strick, J. Allemand, V. Croquette, and D. Bensimon, Twisting and stretching single DNA molecules, Prog. Biophys. Mol. Biol., 74(2000), pp. 115–140.

    Article  CAS  PubMed  Google Scholar 

  131. T.R. Strick, J.F. Allemand, D. Bensimon, A. Bensimon, and V. Croquette, The elasticity of a single supercoiled DNA molecule, Science, 271(1996), pp. 1835–1837.

    Article  CAS  PubMed  Google Scholar 

  132. D. Swigon, Configurations with Self-Contact in the Theory of the Elastic Rod Model for DNA, Doctoral Dissertation, Rutgers University, 1999.

    Google Scholar 

  133. D. Swigon, B. Coleman, and I. Tobias, The elastic rod model for DNA and its application to the tertiary structure of DNA minicircles in mononucleosomes, Biophys. J., 74(1998), pp. 2515–2530.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  134. D. Swigon, B.D. Coleman, and W.K. Olson, Modeling the Lac repressoroperator assembly: the influence of DNA looping on Lac repressor conformation, Proc. Natl. Acad. Sci. U.S.A., 103(2006), pp. 9879–9884.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  135. D. Swigon and W.K. Olson, Mesoscale modeling of multi-protein-DNA assemblies: the role of the catabolic activator protein in Lac-repressor-mediated looping, submitted to Int. J. Nonlin. Mech. (2007).

    Google Scholar 

  136. J.M.T. Thompson and A.R. Champneys, From helix to localized writhing in the torsional post-buckling of elastic rods, in Localization and solitary waves in solid mechanics, Vol. 12 of Adv. Ser. Nonlinear Dynam., World Sci. Publ., River Edge, NJ, 1999, pp. 111–132.

    Google Scholar 

  137. J.M.T. Thompson, G.H.M. Van der HEIJDEN, AND S. Neukirch, Supercoiling of DNA plasmids: mechanics of the generalized ply, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 458(2002), pp. 959–985.

    Article  CAS  Google Scholar 

  138. A.V. Tkachenko, Electrostatic effects in DNA stretching, Phys. Rev. E Stat. Nonlin. Soft. Matter. Phys., 74(2006), p. 041801.

    Article  PubMed  CAS  Google Scholar 

  139. I. Tobias, B.D. Coleman, and W.K. Olson, The dependence of DNA tertiary structure on end conditions: Theory and implications for topological transitions, J. Chem. Phys., 101(1994), pp. 10990–10996.

    Article  CAS  Google Scholar 

  140. I. Tobias, D. Swigon, and B.D. Coleman, Elastic stability of DNA configurations. I. General theory, Phys. Rev. E (3), 61(2000), pp. 747–758.

    Article  CAS  Google Scholar 

  141. M. Vazquez and D.W. Sumners, Tangle analysis of Gin site-specific recombination, Math. Proc. Cambridge Philos. Soc., 136(2004), pp. 565–582.

    Article  Google Scholar 

  142. E. Villa, A. Balaeff, L. Mahadevan, and K. Schulten, Multiscale method for simulating protein-DNA complexes, Multiscale Model. Simul., 2(2004), pp. 527–553 (electronic).

    CAS  Google Scholar 

  143. A. Vologodskii and N. Cozzarelli, Conformational and thermodynamic properties of supercoiled DNA, Annu Rev Biophys Biomol Struct, 23(1994), pp. 609–643.

    Article  CAS  PubMed  Google Scholar 

  144. A. Vologodskii and N. Cozzarelli, Modeling of long-range electrostatic interactions in DNA, Biopolymers, 35(1995), pp. 289–296.

    Article  CAS  PubMed  Google Scholar 

  145. A.V. Vologodskii, W. Zhang, V.V. Rybenkov, A.A. Podtelezhnikov, D. Subramanian, J.D. Griffith, and N.R. Cozzarelli, Mechanism of topology simplification by type II DNA topoisomerases, Proc. Natl. Acad. Sci. U.S.A., 98(2001), pp. 3045–3049.

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  146. J. Wang, DNA topoisomerases, Annu. Rev. Biochem., 65(1996), pp. 635–692.

    Article  CAS  PubMed  Google Scholar 

  147. J.D. Watson and F.H. Crick, Molecular structure of nucleic acids; a structure for deoxyribose nucleic acid, Nature, 171(1953), pp. 737–738.

    Article  CAS  PubMed  Google Scholar 

  148. T.P. Westcott, I. Tobias, and W.K. Olson, Modeling self-contact forces in the elastic theory of DNA supercoiling, J. Chem. Phys., 107(1997), pp. 3967–3980.

    Article  CAS  Google Scholar 

  149. J.H. White, Self-linking and the Gauss integral in higher dimensions, Amer. J. Math., 91(1969), pp. 693–728.

    Article  Google Scholar 

  150. J.H. White, An introduction to the geometry and topology of DNA structure, in Mathematical methods for DNA sequences, CRC, Boca Raton, FL, 1989, pp. 225–253.

    Google Scholar 

  151. P. Wiggins, R. Phillips, and P. Nelson, Exact theory of kinkable elastic polymers, Phys. Rev. E Stat. Nonlin. Soft. Matter. Phys., 71(2005), p. 021909.

    Article  PubMed  PubMed Central  CAS  Google Scholar 

  152. J. Yan, M.O. Magnasco, and J.F. Marko, A kinetic proofreading mechanism for disentanglement of DNA by topoisomerases, Nature, 401(1999), pp. 932–935.

    Article  CAS  PubMed  Google Scholar 

  153. J. Yan and J. Marko, Effects of DNA-distorting proteins on DNA elastic response, Phys. Rev. E Stat. Nonlin. Soft. Matter. Phys., 68(2003), p. 011905.

    Article  PubMed  CAS  Google Scholar 

  154. J. Yan and J. Marko, Localized single-stranded bubble mechanism for cyclization of short double helix DNA, Phys. Rev. Lett., 93(2004), p. 108108.

    Google Scholar 

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

    Article  Google Scholar 

  156. Y. Zhang and D.M. Crothers, Statistical mechanics of sequence-dependent circular DNA and its application for DNA cyclization, Biophys. J., 84(2003), pp. 136–153

    Article  CAS  PubMed  PubMed Central  Google Scholar 

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Acknowledgments

The author wishes to express his thanks to Zuzana Swigonova for careful proofreading of the manuscript and numerous suggestions. Much of this work was written during a stimulating one semester visit at the Institute for Mathematics and its Applications, University of Minnesota. Support by A.P. Sloan Fellowship and NSF grant DMS-05-16646 is also acknowledged.

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  1. Department of Mathematics, University of Pittsburgh, The work was supported by Institute for Math-ematics and its Applications (IMA), Alfred P SLoan Fellowship and NSF Grant DMS 0516646, 301 Thackeray Hall, 15260, Pittsburgh, PA, USA

    David Swigon

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Editors and Affiliations

  1. Dept. Pharmacology & Genome Center, University of California, Davis, E. Health Sciences Drive 451, Davis, 95616, USA

    Craig John Benham

  2. Dept. Chemistry & Biochemistry, Georgia Institute of Technology, Atlanta, 30332-0400, USA

    Stephen Harvey

  3. Dept. Chemistry & Chemical Biology, Rutgers State University of New Jersey, Taylor Rd. 610, Piscataway, 08854-8087, USA

    Wilma K. Olson

  4. Dept. Mathematics, Florida State University, Tallahassee, 32306-4510, USA

    De Witt Sumners

  5. Dept. Mathematics, University of Pittsburgh, Thackeray Hall 301, Pittsburgh, 15260, USA

    David Swigon

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Swigon, D. (2009). The Mathematics of DNA Structure, Mechanics, and Dynamics. In: Benham, C., Harvey, S., Olson, W., Sumners, D., Swigon, D. (eds) Mathematics of DNA Structure, Function and Interactions. The IMA Volumes in Mathematics and its Applications, vol 150. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0670-0_14

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