Phyllotaxis and patterns on plants

<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plantsAlan C. Newell, Zhiying Sun, Patrick D. ShipmanUniversity of ArizonaMarch 23, 2009Alan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

Patterns Are EverywhereAlan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

More Examples of Patterns on PlantsAlan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

How does leaves/flowers form on a plant?Maturesunflower headApical meristem(AM)Schematic of AMAlan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

Diagrams of phylla formationAlan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants



Plant <strong>patterns</strong>Challenges:Why Fibonacci?Connection between phyllotactic configurations <strong>and</strong> surfacedeformationsTransitionsUniversality, self similarityAre plant <strong>patterns</strong> seen anywhere else in the physical world orthe laboratory?Alan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

Mechanisms <strong>and</strong> modelsParadigms: Hofmeister (1868),Snow <strong>and</strong> Snow(1950’s)Douady <strong>and</strong> Couder (1990’s)Mechanical stresses:Green, Steele, Dumais (1990’s)Shipman <strong>and</strong> Newell (2005)Biochemical agents: Reinhardt et al (2000)Meyerowitz et al (2007)Traas et al (2007)Coupled model:Growth affects stress/strainStress promotes growth.How do we connect paradigms <strong>and</strong> mechanisms?Alan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

Paradigms <strong>and</strong> Mechanisms: connections?Paradigms are rules for the placement of the newest primordiumgiven the positions of all previous primordia.cf. packing efficiencyMechanisms involve instabilities of uniform states, the decompositionof the phase space into active A <strong>and</strong> passive P(A) modes, thecoordinatization of A with (amplitude) order parameters, <strong>and</strong> orderparameter equations which may be gradient flows with a free energy.cf. energy minimizationHow to connect these ideas?Alan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

Biochemical factorsReinhardt (2000) showed that hormone Auxin is instrumental ingrowth <strong>and</strong> uneven distribution of Auxin is the reason for theformation of phyllotaxisAlan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

Biochemical factorsReinhardt (2000) showed that hormone Auxin is instrumental ingrowth <strong>and</strong> uneven distribution of Auxin is the reason for theformation of phyllotaxisMeyerowitz, Traas, Reinhardt et al (2006): Auxin efflux proteinPIN1 on the cell wall can redistribute themselves so as to pumpAuxin form low concentration to high concentrationdA(i)dt= D ∑ k(A(k) − A(i)) + T ∑ k(A(k)P(k, i) − A(i)P(i, k)) + c − dA(i)P(i, j) = P ij = PA(j)κ + ∑ (k)A(k) jA(i): Auxin concentration in cell iP(i, k): PIN1 concentration on cell wall (i,j) pointing to cell jAlan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

Biochemical factorsReinhardt (2000) showed that hormone Auxin is instrumental ingrowth <strong>and</strong> uneven distribution of Auxin is the reason for theformation of phyllotaxisMeyerowitz, Traas, Reinhardt et al (2006): Auxin efflux proteinPIN1 on the cell wall can redistribute themselves so as to pumpAuxin form low concentration to high concentrationdA(i)dt= D ∑ k(A(k) − A(i)) + T ∑ k(A(k)P(k, i) − A(i)P(i, k)) + c − dA(i)P(i, j) = P ij = PA(j)κ + ∑ (k)A(k) jA(i): Auxin concentration in cell iP(i, k): PIN1 concentration on cell wall (i,j) pointing to cell jThe continuum limit PDE:g t = −Lg −H ▽ 2 g −▽ 4 g − ¯κ 1 ▽(g ▽g)− ¯κ 2 ▽(▽g ▽ 2 g)−δ ′ g 3g(r, α): auxin fluctuation around stationary uniform concentration∼ growth strainAlan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

Mechanical FactorsFvKD Equation(Describing deformation of thin elastic shells):ζw t + D∆ 2 w + N∆ χw + 1R α∆ ρf − [f , w] + κw + γw 3 = 01Eh ∆2 f − 1R α∆ ρw + 1 [w, w] = 02w(r, α): out-of-surface deformation. f (r, α): in-plane stress.Alan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

Mechanical FactorsFvKD Equation(Describing deformation of thin elastic shells):ζw t + D∆ 2 w + N∆ χw + 1R α∆ ρf − [f , w] + κw + γw 3 = 01Eh ∆2 f − 1R α∆ ρw + 1 [w, w] = 02w(r, α): out-of-surface deformation. f (r, α): in-plane stress.Combined ModelAlan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

Equations for coupled modelw(r, α, t)f (r, α, t)g(r, α, t)Surface normal deformationAiry Stress tensorGrowth/fluctuation auxin concentrationζw t + D∆ 2 w + N∆ χ w + 1R α∆ ρ f − [f , w] + κw + γw 3 = 01Eh ∆2 f − 1R α∆ ρ w + 1 2[w, w]+∆g = 0g t = −Lg − H ▽ 2 g − ▽ 4 g − ¯κ 1 ▽ (g ▽ g) − ¯κ 2 ▽ (▽g ▽ 2 g) −δg 3 + β∆fAlan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

Patterns arising from the governing equationsBoth mechanisms share the same structures in the equations.∂w∂t= −(∇ 2 + P) 2 w + ɛw + N.L..In 1-dimensional, The linear growth rate for e ikx+σt isσ(k) = (−k 2 + P) 2 + ɛ.In 2-dimensional, ∇ 2 = ∂2 + 1 ∂∂r 2 r ∂r + 1 ∂ 2The linear growth rate forr 2 ∂α 2e ilα+imx+σt is σ(l, m) = −(−l 2 − m2 + P) 2 + ɛ.r 2Triad: ⃗ k m = (l m , m), ⃗ k m + ⃗ k n = ⃗ k m+n ⇒ l m + l n = l m+n , m + n = (m + n)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

Amplitude Equations 1. Representationw(r, α, t) = ∑ A s m(r, t)e is ∫ l mdr−imα + (∗)where A m (r, t)e i ∫ l mdr is WKB approximation for Hankel functionsH s m(kr) in the r >> 1, m/r finite limit with l 2 m = k 2 − m 2 /r 2g(r, α, t) = ∑ B s m(r, t)e is ∫ l mdr−imα + (∗)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

Amplitude Equations 1. EquationsLinear∂A m∂t∂+ (2il m∂r + i ∂l m+ i l m∂r r )2 A m = (ɛ − (lm 2 + m2r 2 − P)2 )A mQuadratic nonlinearitye i ∫ l mdr−imα e i ∫ l ndr−inα → e i ∫ (l m+l n)dr−i(m+n)α+τ(m, n, m + n)A ∗ nA m+nCubic Saturation (e.g. elastic foundation response)−γA m (|A m | 2 + ∑ δ ms |A s | 2 )Alan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

Energy FunctionalE{A m , A m ∗, ...} = − ∑ σ m A m A ∗ m−∑ τ jpq (A ∗ j A pA q + (∗))+ γ ∑ ( 1 2 |A m| 4 + ∑ m≠sδ ms |A m | 2 |A s | 2 )Adiabatic responseSolve stationary amplitude equationsChoose l m , l n to minimize EA j (l m , l n , r)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

OutcomeTheorem: The amplitude equations for ANY Fibonacci sequence areinvariant under R → Rφ, φ = √ 5+12, the golden number;A s j(R) → A−sj+1 (Rφ), l j(R) → −l j+1 (Rφ), m j → m j+1 (≃ m j φ)1,2,3,5,8,13,...Alan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

PDE Simulationr = r 0 + Ut − r ′∇ 2 = 1 R∂∂r ′ R ∂∂r ′ + 1 R 2 ∂ 2∂α 2 R = r 0 + ( U V − 1)r ′Alan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

PDE SimulationFront propagating outwards (radius increasing R ∈ [2, 7],α ∈ [0, 2π])Fourier Space:(Frequency)Alan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants




















PDE SimulationAlan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

OutcomeTheorem: Under R → Rφ mn , φ mn = m+nφmφ+n = 1+φ22φ≃ φ − 1/2, theamplitude equations of two different Fibonacci sequences areisomorphic <strong>and</strong> in particular,E(1, 2, 3, 5, 8, ...; R) = E(1, 3, 4, 7, ...; 1.17R)1,3,4,7,...Alan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

TransitionsHow are transitions achieved between whorls <strong>and</strong> Fibonaccisequences <strong>and</strong> between different Fibonacci sequences?e.g. 2,2,4 → 2,3,5e.g. 1,2,3,5, 8 → 1,3,4,7,..cf. Eckhaus, skew-varicose instabilitiesAlan C. Newell, Zhiying Sun, Patrick D. Shipman<strong>Phyllotaxis</strong> <strong>and</strong> <strong>patterns</strong> on plants

Phyllotaxis and patterns on plants

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