Fig 1.
Initial domain and growth pattern.
(A) Initial domain: a cylinder of radius R = 10 and height Hz = 6, with a hemispherical tip. (B)-(D) Growth patterns of the domain (tip growth). The Gaussian distribution G[u] has a width of R/2 and it can be seen at the tip in (B) and (C). Images are taken at times T = 450, 550, 650.
Fig 2.
Whorled phyllotactic patterns.
Time sequence of emergence of primordia into whorled phyllotactic patterns. Parameter values are β = 0.5, γ = 0.2, c = 0.57, G[u] with a width of R/2, κ = 2, and the rest of parameters as indicated in the text. The whorls appear in the regions where the mean curvature changes more abruptly. (A) Four-whorled phyllotactic pattern, η = 0.3902. Images are taken at times T = 150, 250, 350, 450. (B) Three-whorled phyllotactic pattern, η = 0.2601. Images are taken at times T = 350, 450, 550, 650. (C) Mean curvature of the domain at times T = 150 and T = 350 for the 4-whorled pattern evolution shown in (A). (D)-(E) Top view of (A) and (B) at times T = 350, and T = 550, respectively.
Fig 3.
Patterns formed with no stress or spontaneous curvature. (A) Time evolution of the pattern shown in Fig 2A, but with values β = 0, γ = 0. (B) Same as (A), but only β = 0 (γ = 0.2 as in Fig 2A). Images are taken at times T = 150, 250, 350, 450.
Fig 4.
Patterns formed with high stress and a wider Gaussian distribution G[u]. (A) Time evolution of the pattern shown in Fig 2A, but with γ = 3.5. Compare the top right panel, taken at time T = 650, with Fig 2B. (B) Same as Fig 2A, but the width of G[u] is R/2.5.
Fig 5.
Phase diagram for whorled patterning.
Whorled phyllotactic patterns emerge as a function of the stress tensor coupling. When γ increases the emergence of primordia is delayed and the whorls become distorted. The symmetry of whorled patterns increases as as function of η. The dotted lines represent the boundary limits for 3- to 7-symmetry (S3, S4, S5, S6, S7) and the numerical limit values are indicated in red on η axis. Note that in some cases, the increment of γ maintains the symmetry (S3, S4, S5, S6) but it augments the symmetry in other cases (S5 → S6, S6 → S7). The dash blue lines delimitate the whorled and aberrant patterns (AB). The values of η for each image are indicated on axis (in black) and the rest of the parameter values are β = 0.5, G[u] with a width of R/2, κ = 2 and c = 0.33 for η = 0.65 and η = 0.89; and c = 0.57 for η = 0.26 and η = 0.39.
Fig 6.
Phase diagram for 3-symmetry whorled patterning.
Whorled phyllotactic patterns change with spontaneous curvature interaction β and stress tensor coupling γ. When γ increases β decreases in order to maintain the emergence of primordia as whorled patterns. Aberrant patterns (AB) emerge outside of the dash blue lines. Parameter values are η = 0.26, G[u] with a width of R/2, κ = 2 and c = 0.57.
Fig 7.
Time sequence of emergence of spiraling patterns. Parameter values are β = 0.5, γ = 0.2 and the rest of parameters as indicated in the text. (A) Spiral pattern with 5-fold symmetry, η = 0.5676, κ = 2, c = 0 and G[u] with a width of R/2. Images are taken at indicated times. (B) Ribs pattern with 6-fold symmetry, η = 0.6504, κ = 0.5, c = 0 and G[u] with a width of R/1.5,. Images are taken at indicated times. (C) Five-whorled phyllotactic pattern, η = 0.8437, κ = 2, c = 0.33 and G[u] with a width of R/2. Packing of primordia into parastichies can be seen at time iteration T = 500 and on.
Fig 8.
Phase diagram for ribbed patterning.
Ribbed patterns emerge as a function of the stress tensor coupling. The symmetry of ribs increases as a function of η. The dotted lines represent the boundary limits for 3- to 8-symmetry (S3, S4, S5, S6, S7, S8) and the numerical limit values are indicated in red on η axis. Note that in all cases the symmetry of the pattern is below of the limit value, which means that in some cases, the stress coupling γ raises the symmetry. Emergence of aberrant patterns (AB) is indicated with dashed blue lines. The values of η for each image are indicated on η axis (in black) and the rest of the parameter values are β = 0.5, G[u] with a width of R/1.5, κ = 0.5 and c = 0.
Fig 9.
(a) Three-whorled phyllotactic pattern as in Fig 2(e); the different floral organs emerge in time as whorls of three. (b) Phyllotactic pattern of ribs with 6-symmetry as in Fig 7(b). (c) Phyllotaxis of Pachypodium lamerei; plants of this genus exhibit the pattern shown in Fig 7(c): their leaves first emerge as whorls and then pack into parastichies.
Fig 10.
Mechanical model without the interaction with the chemical model.
Shapes obtained starting with an initial hemisphere defining ϕ = 0, and using various stationary forms for the field u. (A) Field u is a Gaussian centered at the symmetry axis with increasing width. From top to bottom, width is 0, R/4, R/2 and R. (B) Field u on the surface follows a fixed sinusoidal vertical pattern. (C) A two dimensional u fixed field with maxima at the tip and at the bottom of the initial domain. (D) Gaussian distribution of u with center off the symmetry axis. There is no Turing mechanism in (A), (B) and (C).