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Dynamics of Twist and Writhe and the Modeling of Bacterial Fibers

  • Michael Tabor
  • Isaac Klapper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 82)

Abstract

We discuss a range of issues associated with the dynamics of twist and writhe including some new theoretical and numerical results and techniques. A precise understanding of twist and writhe is important in a variety of physical and biological processes and, in particular, we describe how these ideas can be used to model the dynamics of the self-assembling bacterial fiber, bacilus subtilis.

Keywords

Contact Force Space Curve Vortex Filament Nonlinear Schrodinger Equation Twist Rate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Do Carmo, M.P. (1976), Differential Geometry of Curves and Surfaces (Prentice-Hall, Englewood Cliffs, NJ).Google Scholar
  2. [2]
    Hasimoto, H. (1972), A soliton on a vortex filament, J. Fluid Mech. 51, 477–485.CrossRefGoogle Scholar
  3. [3]
    Lamb, G.L. (1977), Solitons on moving space curves, J. Math. Phys. 18, 1654–1661.CrossRefGoogle Scholar
  4. [4]
    Langer, J. and Perline, R. (1991), Poisson geometry of the filament equations, J. Nonlinear Sci. 1, 71–93.CrossRefGoogle Scholar
  5. [5]
    Keener, J.P. (1990), Knotted vortex filaments in an ideal fluid, J. Fluid Mech. 211, 629–651.CrossRefGoogle Scholar
  6. [6]
    Goldstein, R.E. and Petrich, D.M. (1991), The Korteweg de Vries hierarchy as dynamics of closed curves in the plane, Phys. Rev. Lett. 67, 3203–3206.CrossRefGoogle Scholar
  7. [7]
    Berry, M.V. and Hannay, J.H. (1988), Classical non-adiabatic angles, J. Phys. A: Math. Gen. 21, L325–L331.CrossRefGoogle Scholar
  8. [8]
    Rolfsen, D. (1976), Knots and Links (Publish or Perish, Berkeley, CA).Google Scholar
  9. [9]
    Gauss, C.F. (1877), Zur mathematischen théorie der electrodynamischen Wirkungen, Koniglichen Gesellschaft der Wissienshaften zu Gottingen 5, 602–629.Google Scholar
  10. [10]
    Milnor, J.W. (1965), Topology from the Differentaible Viewpoint (The University Press of Virginia, Charlottesville).Google Scholar
  11. [11]
    Pohl, W.F. (1980), DNA and differential geometry, Math. Intelligencer 3, 20–27.CrossRefGoogle Scholar
  12. [12]
    Berger, M.A. and Field, G.B. (1984), The topological properties of magnetic helicity, J. Fluid Mech. 147, 133–148.CrossRefGoogle Scholar
  13. [13]
    Moffatt, H.K. and Ricca, R.L. (1992), Helicity and the Calugareanu invariant, Proc. Roy. Soc. A 439, 411.CrossRefGoogle Scholar
  14. [14]
    Zajac, E.E. (1962), Stability of two planar loop elasticas, J. Appl. Mech. 29, 136–142.Google Scholar
  15. [15]
    Benham, C.J. (1989), Onset of writhing in circular elastic polymers, Phys. Rev. A 39, 2582–2586.CrossRefGoogle Scholar
  16. [16]
    Fuller, F.B. (1971), The writhing number of a space curve, Proc. Natl. Acad. Sci. USA 68, 815–819.CrossRefGoogle Scholar
  17. [17]
    Fuller, F.B. (1978), Decomposition of the linking number of a closed ribbon: A problem from molecular biology, Proc. Natl. Acad. Sci. USA 75, 3557–3561.CrossRefGoogle Scholar
  18. [18]
    Aldinger, J., Klapper, I., and Tabor, M. (1995), Formulae for the calculation and estimation of writhe, J. Knot Theory Ramifications 4, 343–372.CrossRefGoogle Scholar
  19. [19]
    Klapper, I. and Tabor, M. (1994), A new twist in the kinematics and elastic dynamics of curves and ribbons, J. Phys. A: Math. Gen. 27, 4919–4924.CrossRefGoogle Scholar
  20. [20]
    Mendelson, N.H. (1990), Bacterial macrofibers: the morphogenesis of complex multicellular bacterial forms, Sci. Progress Oxford 74, 425–441.Google Scholar
  21. [21]
    Thwaites, J.J. and Mendelson, N.H. (1991), Mechanical behavior of bacterial cell walls, Adv. Microbiol. Physiol. 32, 174–222.Google Scholar
  22. [22]
    Love, A.E.H. (1927), A Treaty on the Mathematical Theory of Elasticity, Fourth Edition (Cambridge University Press, Cambridge, reprinted by Dover Publications, New York).Google Scholar
  23. [23]
    Landau, L.D. and Lifschitz, E.M. (1959), Theory of Elasticity (Pergamon Press, Oxford).Google Scholar
  24. [24]
    Coleman, B.D., Dill, E.H., Lembo, M., Lu, Z., and Tobias, I. (1993), On the dynamics of rods in the theory of Kirchhoffand Clebsch, Arch. Rational Mech. Anal. 121, 339.CrossRefGoogle Scholar
  25. [25]
    Simo, J.D., Marsden, J.E., and Krishnaprasad, P.S. (1988), The Hamiltonian structure of nonlinear elasticity: the material and convective representations of solids, rods and plates, Arch. Rational Mech. Anal. 104, 125–183.CrossRefGoogle Scholar
  26. [26]
    Maddocks, J.H., et al., this volume.Google Scholar
  27. [27]
    Schlick, T. and Olson, W.K. (1992), Trefoil knotting by molecular dynamics simulations of supercoiled DNA, Science 257, 1110–1115.CrossRefGoogle Scholar
  28. [28]
    Klapper, I. (1994), Biological applications of the dynamics of twisted elastic rods, J. Comp. Phys. in press.Google Scholar
  29. [29]
    Langer, J. and Singer, D.A. (1985), Curve straightening and a minimax argument for closed elastic curves, Topology 24, 75–88.Google Scholar
  30. [30]
    Shi, Y. and Hearst, J.E. (1994), The Kirchhoff elastic rod, the nonlinear Schrödinger equation, and DNA supercoiling, to appear, J. Chem. Phys. Google Scholar
  31. [31]
    Tsuru, H. and Wadati, M. (1986), Elastic model of highly supercoiled DNA, Biopolymers 25, 2083–2096.CrossRefGoogle Scholar
  32. [32]
    Mendelson, N.H. and Thwaites, J.J. (1990), Bending, folding and buckling processes during bacterial macrofiber morphogenesis, Mat. Res. Soc. Symp. 174, 171–178.CrossRefGoogle Scholar
  33. [33]
    Mendelson, N.H. (1976), Helical growth of Bacillus subtilis: A new model of cell growth, Proc. Natl. Acad. Sci. 73, 1740–1744.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • Michael Tabor
    • 1
  • Isaac Klapper
    • 2
  1. 1.Program in Applied MathematicsUniversity of ArizonaTucsonUSA
  2. 2.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

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