On the curving and twining of stems

https://doi.org/10.1016/0098-8472(89)90042-7Get rights and content

Abstract

Spatial and temporal patterns of curvature in seedling stems and an equation relating change of curvature to local growth rates are reviewed. Then twinning growth is analysed. In the basal two-thirds of the growth zone of Pharbitis nil, tissue elements elongate as they are displaced along a helical trajectory. The form of the plant is steady in a stationary reference frame. The stem of morning glory is therefore treated as a tube of radius ω/2 surrounding a generative helix, α. The position vector to points on or in the stem is given by p(s, ϖ, r) = α(s) + r(n cos ϖ + b sin ϖ) where s is arc length along α, r is distance from α(s), ϖ is measured in radians from the normal to α(s), and n and b are the unit normal and binormal to α(s). Lengths and displacement velocities of elements on or in the stem are calculated using the metric tensor for the curvilinear, non-orthogonal s, ϖ, r coordinate system. Then, a form of the strain rate tensor is used to compute elongation rates of line elements in the stem. The steady helical growth pattern can be produced with a growth rate distribution which is rather uniform through the cross-section. Tapered stems elongate slightly more rapidly on the side away from the support.

References (26)

  • W.K. Silk et al.

    Kinematics of plant growth

    J. theor. Biol.

    (1979)
  • L. Baillaud

    Les mouvements d'exploration et enroulement des plantes volubiles

  • M. Boas

    Mathematical methods in the physical sciences

    (1966)
  • C. Darwin

    The movements and habits of climbing plants

    (1876)
  • M.P. Do Carmo

    Differential geometry of curves and surfaces

    (1976)
  • H.E. Dolk

    Ueber die Wirkung der Schwerkraft auf Koleoptilen von Avena sativa—II

  • R.O. Erickson et al.

    An analysis of root growth in cellular and biochemical terms

    Growth

    (1951)
  • R.D. Firn et al.

    The establishment of tropic curvatures in plants

    Ann. Rev. Pl. Physiol.

    (1980)
  • V. Gaba et al.

    The control of cell growth by light

  • P.W. Gandar

    Growth in root apices—I. The kinematic description of growth

    Bot. Gaz.

    (1983)
  • C.R. Goodall et al.

    Quantitative analysis of surface growth

    Bot. Gaz.

    (1986)
  • P.B. Green

    Growth and cell pattern formation on an axis. Critique of concepts, terminology and modes of study

    Bot. Gaz.

    (1976)
  • P.B. Green

    Analysis of axis extension

  • Cited by (27)

    • Morphoelastic rods III: Differential growth and curvature generation in elastic filaments

      2020, Journal of the Mechanics and Physics of Solids
      Citation Excerpt :

      For instance, it is well known that inclining a plant will produce a gravitropic response where part of the stem will grow faster as to recover ascension against gravity by locally creating curvature (Bastien et al., 2013; Kutschera, 2001; O’Reilly and Tresierras, 2011). Similarly, for a twining vine to grow helically around a pole, different points in the section normal to the axis must grow at different rates (Silk, 1989a; 1989b) in order to generate both torsion and curvature. Another interesting case of differential growth is found in the main trunk of some trees that grow straight but twisted (Schulgasser and Witztum, 2006).

    • Attachment and interfacial strength between twining plants and the support

      2017, Extreme Mechanics Letters
      Citation Excerpt :

      They will get a larger steady wavelength concomitant with smaller deformation when replacing the support for a thicker one, but further loss the ability to ascend a support when the radius of the support exceeds a critical value [7,8,10]. The existing analytical theories proposed by most authors are based on helical rope or spring model [11–13], where, by simplifying the Euler relation developed mostly in belt conveyor, the core notion concentrates on the friction provided by the tension conducting squeezing force between the plants and support to avoid the slippage, which also indicates a surprisingly exponential increasing of tension force along the twining stem. However, such strong contraction is not observed when we cut off the twining stem from its straight shoot.

    • Morphoelastic rods. Part I: A single growing elastic rod

      2013, Journal of the Mechanics and Physics of Solids
      Citation Excerpt :

      Understanding the growth, formation and dynamics of these fundamental structures is not only of intrinsic theoretical interest, but it also lies at the heart of a host of important processes in biology, physics, and engineering (Zajac, 1962; Barkley and Zimm, 1979; Spruit, 1981; Benham, 1979, 1983; Keener, 1990; DaSilva and Chouduri, 1993; Manning et al., 1996; Thompson et al., 2002). The main motivation for the research presented here is the fascinating growth of various biofilaments as observed in bacterial fibres (Mendelson, 1978; Klapper and Tabor, 1996; Goriely and Tabor, 2000; Goriely et al., 2008), bacterial filaments (Gray et al., 1990; Shapiro and Dworkin, 1997; Goriely and Tabor, 2003a,b), fungi (Robertson, 1968; Bergman et al., 1969; Koch, 1994; Goriely and Tabor, 2011), root hairs (Dumais et al., 2004; Roelofsen and Houwink, 1953; Shaw et al., 2000), stems (Silk and Haidar, 1986; Silk, 1989; Steele, 2000; Goriely and Neukirch, 2006; Moulia et al., 2006; Speck and Burgert, 2011), roots (Chavarrıa-Krauser et al., 2005; Erickson and Sax, 1956; Mullen et al., 1998; Okada and Shimura, 1990; Buschmann et al., 2009), tendrils (Keller, 1980; Goriely and Tabor, 1998; McMillen and Goriely, 2002; Goldstein and Goriely, 2006), neurons (Barlow et al., 1989; Dennerll et al., 1989; Lamoureux et al., 1989, 1992, 2002; Chada et al., 1997), umbilical cords (Goriely, 2004; Miller et al., 1982), tendons (Rao et al., 2003), arteries (Alford et al., 2007; Taber and Humphrey, 2001; Goriely and Vandiver, 2010), and the spine (Entov, 1983), to name but a few. On the mechanical side, there is a considerable body of work and current interest in volumetric three-dimensional growth theories, dating back to Skalak and Hoger's seminal work (Skalak, 1981; Skalak et al., 1982; Rodriguez et al., 1994) as well as Stein (1995) and Rachev (1997).

    • Modelling of root growth and bending in two dimensions

      1997, Journal of Theoretical Biology
    View all citing articles on Scopus
    View full text