Images for Nonlinear Dynamics of Passionflower Tendril Free Coiling Christina Cogdell and Paul Riechers NCASO Spring 2012
Figure 1: Typical Free Coiling Patterns Showing Both Structure and Randomness Cylindrical Conical Reversals Knots Multiple Perversions 90-degree Single Perversion
Figures 2-3 plus bonus images: Existing Tendril Coiling Models - Invariant Helical Contact Coils Explained Using Mechanical Models Based upon Kirchhoff s Equations for Rods with Intrinsic Curvature at Equilibria in Minimal Energy State From Alain Goriely and Michael Tabor, Spontaneous Helix Hand Reversal and Tendril Perversion in Climbing Plants, Physical Review Letters 80:7 (16 Feb. 1998): 1564-67. Kirchhoff s equations are used to explain the presence of one or more perversions in a coil that is fixed at both ends. Actual Joined Passionflower tendils, one showing conical shape and one (below) approximating the form imagined for the Kirchhoff s equation models. Far right: Two contact coils joined together ratcheting down.
Figure 4: Diameter of Coil 2 =26.17% D =46.06% d =27.77% d = 0-2.99 mm D = 3-7.99 mm 2 = > 8mm Shannon Entropy L5 H(D) = 5.33 bits First Order Markov Chain for Diameter This maps the probability of seeing any subsymbol given knowledge of only the preceding subsymbol. d D 2 d D 2
Figure 5. Periodicity (# of Coils/5mm) 3 =19.39% p =53.32% P =27.29% p = 0-1 coil P = 1-4 coils 3 = > 4 coils p P 3 Shannon Entropy L5 H(P) = 4.10 bits First Order Markov Chain for Periodicity
Figure 8. Angular Axis Rotation 4 =76.57% 9 =15.14% 4 = 0-44.99 degrees 9 = 45-90 degrees 8 = 180 degree reversal 8 =8.29% Shannon Entropy L5 H(A) = 3.8 bits 4 9 9 First Order Markov Chain for Angular Axis Rotation 8 8
Figure 9. Contact Status f =94.45% c =5.55% c = self-contact f = free f Shannon Entropy L5 H(C) = 0.85 bits First Order Markov Chain for Contact Status c c 0503 6:30am 0504 7:30am 0505 11:00am front 0506 8:00pm side view 0507 10:00am side view Final, bottom side view
Figures 10 and 11: Free Coils with Perversions and Coiling in Multiple Places at Once Figure 10: Coiling with a Perversion
Figure 12: G-fiber action in G-fiber cells causes contraction and twisting. For bidirectional coiling tendrils, there is a cylinder of G-fiber cells around the tendril, responsive to touch from any side. Only a portion on the contact side become active, whereas auxin causes cell elongation on the side opposite contact.
Figure 13: Hypothetical Model of the Process of Tendril Coiling: G-Fiber Contraction on Contact Side + Oppositional Auxin Gradient-Induced Cell Elongation Contraction and Twist from G-Fiber Action on Contact Side (Concave Side) Contraction Mechanical Stress Travels Longitudinally Up and Down the Concave Side from G- Fiber Contraction + Differential Lignification for Variable Stiffening Cell Elongation from High Auxin Levels on Side Opposite Contact (Convex Side) Elongation Mechanical Stress Travels Longitudinally Up and Down the Convex Side from Auxin-Induced Elongation
Figure 14-15: Role of the Gelatinous Fiber Layer in Tendril Contraction on the Concave Side of Figure 14. Gelatinous-fiber cells have 3 cell wall layers: Primary, S1 secondary, and S2 secondary, each of which has cellulose microtubules (MTs) that provide structural support. The alternating orientation of MTs is key to cell deformation patterns under G-fiber action. Figure 15, below: Multilayered structure of the g-fiber cell, from Yamamoto, 2004.
Figure 16, left: Diagram of mechanical principles of coiling showing abstracted tendril cylinder with cross sections under twist + bend action, which results in a uniform helix. Diagram by Chun-Feng Liu, Emergent Technologies and Design, Architectural Association. How to Make an Invariant Helix: Twist + Bend Figure 17: Twist + Bend possible outcomes, using sine and cosine equations, by Paul Riechers. Only helices, circles, arcs and lines of uniform curvature are possible when the twist + bend angle parameters do not change.
Working Hypothesis of Tendril Coiling Process: Auxin Acts as a Morphogen Triggering Cell Elongation on Convex Side of Coil Figure 18, above: Phototropic response showing PIN3 polarization carrying auxin to side opposite light toward greatest auxin concentration, where it triggers cell elongation causing the plant to curve and grow toward the light. From Ding, Zhaojun et al, Light-mediated polarization of the PIN3 auxin transporter for the phototropic response in Arabidopsis, Nature Cell Biology 13:4 (April 2011): 447-53. Figure 19, above: Diagram showing auxin gradients and polar auxin transport in Arabidopsis root apex. From Bhalero, Rishikesh and Malcolm Bennett, The case for morphogens in plants, Nature Cell Biology 5:11 (November 2003): 939-42.
Figures 20-21: Auxin s Role in Tropic Responses to Environmental Stimuli: Curvature Variations via Cell Elongation in Phototropism, Gravitropism, Thigmotropism Above: Diagram showing auxin diffusion into a cell and active transport out of the cell via PIN1. From Benjamin, Rene, and Ben Scheres, Auxin: The Looping Star in Plant Development, Annual Review of Plant Biology 59 (2008): 443-65. Above: Alignment of Microtubules and PIN1 polarity orientation owing to upstream mechanical stress; shoot apical meristem epidermal cells respond to stress by reorganizing their MT arrays to be parallel to the direction of the largest principle stress. PIN1 alignment then directs auxin along its polarity path unidirectionally toward the opposite side where cells elongate. From Heiser, Marcus et al, Alignment between PIN1 Polarity and Microtubule Orientation in the Shoot Apical Meristem Reveals a Tight Coupling between Morphogenesis and Auxin Transport, PLOS Biology 8:10 (October 2010): e1000516.
Figure 22: Some of the fascinating patterns that free coiling tendrils exhibit that have changing radii, periodicity and curvature, including perfect spirals.