Fibonacci Patterns in Plants

Transcription

1 Fibonacci Patterns in Plants Matt Pennybacker Alan Newell Zhiying Sun Patrick Shipman The University of Arizona 29 March 2010 Matt Pennybacker, The University of Arizona Fibonacci Patterns 1/12

2 An Example Matt Pennybacker, The University of Arizona Fibonacci Patterns 2/12

3 History Placement of primordia is determined by a set of rules, the most important being: New primordia are formed periodically in the largest available space left by the previous ones. Pursuing this approach leads to the conclusion: The ordering is explained due to the system s trend to avoid rational (periodic) organization, thus leading to a convergence towards the golden mean. Hofmeister (1868); Douady and Couder (1996); Atela, Golé, and Hotton (2002). Matt Pennybacker, The University of Arizona Fibonacci Patterns 3/12

4 for Fibonacci patterns, the choice between various competitors for the influenced by the bias and presence of the previously created pattern on of the new pattern. In this first scenario, the pattern furthest away with the least number of circumferential wavelengths and the circumfe phyllotactic pattern increase towards the apex. We shall learn later the 1 Rapid circumferential mitosis produces wavenumbers, all integers, belong to the Fibonacci sequ undifferentiated cells. The Shoot Apical Meristem 2 Primordia form on the generative annulus. 3 Existing primordia move outward and continue their development. Model Statement Mechanics and biochemistry are the primary contributors to primordium formation. Figure 2: Diagram of the shoot apical meristem. In the regular proc region 1, the pattern forms in region 2 the generative annulus, and the into, for example, leaves as they move out to region 3. Matt Pennybacker, The University of Arizona Fibonacci Patterns 4/12

5 t) w(r Growth/auxin concentration fluctuation limit)tens, α, t) Surface normal deformation, f (r, (continuum α, t) Airy Stress Growth g(r, α, t)mechanisms Growth/auxin concentration fluctuation (continuum limit [Reinhardt, Meyerowitz, Traas e [Reinhardt, Meyerowitz, Traa Mechanics: Primordia appear as buckling of the surface of the SAM, which can be modeled as a hard shell on an elastic foundation. Biochemistry: Primordium locations are mediated by local concentrations +ofn w +fhormone ρ f auxin, [f, w]which + κwis+3γw wt + D ζw wt ++D N wχ w +ther1χαplant = 0= α [f, w] + κw + γw ρ R regulated by the 1in turn 1 1 PIN1 protein. 2 1 f Rα 1 ρ w + 2 [w, w]+ g = 0 2 f Eh w + [w, w]+ g =0 ρ 2 Rennich 2 4(1996); Jönsson, et al (2005). 2 Greene, Steele, and grt α= Lg H g g κ 1 (g g) κ 2 ( g g =Matt Lg H g g κ 1 (g g) κ ( g 5/12 g) Pennybacker, The University of Arizona Fibonacci 2 Patterns22

6 The Model Equations Let w be the surface deformation field, f be the fluctuation of the Airy stress potential, and g be the growth strain field. Model Equations w t = 2 w κw γw 3 P χ w + ν 1 [f, w] C ρ f 2 f C ρ w + (2ν) 1 [w, w] + g = 0 g t = Lg H g 2 g κ 1 (g g) κ 2 ( g g) δg 3 + β f Near the onset of pattern formation, the behavior is governed more by the overall symmetries. We simplify this model to include only the main ingedients. Swift-Hohenberg Equation u t = ( 2 + k 2 0 )2 u + εu + ϑ( u) 2 2ϑ (u u) γu 3 Matt Pennybacker, The University of Arizona Fibonacci Patterns 6/12

7 Aside: Why Hexagons? e i k 1 Another Swift-Hohenberg Equation u t = ( 2 + 1) 2 u + εu + ϑu 2 γu 3 When do k 1, k 2, k 3 = k 1 Examine the the uniform solution to find that modes with ε (k 2 1) 2 > 0 are unstable. When do k 1, k 2, and k 3 = k 1 + k 2 all lie on the unit circle? Quadratic interactions fa Matt Pennybacker, The University of Arizona Fibonacci Patterns 7/12

8 Simulation Results Matt Pennybacker, The University of Arizona Fibonacci Patterns 8/12

9 Local Spectra of Results Matt Pennybacker, The University of Arizona Fibonacci Patterns 9/12

10 Local Spectra of Results Matt Pennybacker, The University of Arizona Fibonacci Patterns 10/12

11 Why Fibonacci? ide of the active set at R = 8. Matt Pennybacker, The University of Arizona Fibonacci Patterns 11/12

12 Thank You Matt Pennybacker, The University of Arizona Fibonacci Patterns 12/12