Fig 1.
Schematic diagram for pentamerous flower development.
Sepal initiation (the first row), arrangement of sepal (black) and petal (white) whorls in blooming flower (the second row). Green circle represents a floral meristem (FM). Index numbers indicate the initiation order of five sepals. The radial position of the organs (the third row), namely the distance between the organ and floral apex, is spaced regularly in a spiral arrangement, whereas it has a gap between the fifth and sixth organs in the pentamerous pseudo-whorled and whorled arrangement. Regarding the hypothetical time evolution of the radial position (the fourth row), in all arrangements, the radial position increases with the progression of floral development. In the spiral arrangement, the radial position of the organ is always spaced regularly. In the pseudo-whorled and whorled arrangement subsequent to helical initiation, the radial position of organs within a whorl becomes closer during growth. In the whorled arrangement following simultaneous initiation, the radial position of the organs within a whorl is always identical.
Fig 2.
Emergence of multiple whorls in model simulations.
A. Geometric assumptions of the model. B. The initiation process. A new primordium (i) is initiated at the edge of the floral meristem (FM; green circle) where the initiation potential Uini takes the minimum value. i, i −1, and i −2 are the primordium indices that denote the initiation order. Uini exponentially decreases with time (α) and the distance between primordia (λini). C. The growth process. Each primordium (k) moves at the outside of the circular FM, depending on the growth potential Ug, k. Primordium k rarely moves against the gradient (grey thin arrow), but mostly follows the gradient (black thick arrow; see the Model section). D–F. Emergence of whorled-type pattern with increasing meristem radius R0 and temporal decay rate α. Left panels: Spatial pattern after 15 primordia (red circles) initiated in an indexed order at the meristem edge (green circle; r = R0). Middle panels: Radial distance (black) from the meristem center as a function of the primordium initiation index (left panel) averaged over 400 replicate Monte Carlo simulations. Error bars represent twice the S.D. Red circles are a set of representative samples. Right panels: Time evolution of the radial coordinates of each primordium averaged over 400 replicates. Error bars show 2 S.D. The arrowheads in D and F indicate the growth arrest of the fifth and sixth primordia, respectively. Colors denote the index of the primordia. Green line in the left, middle and right panels denotes the meristem edge. (R0, α) = (20.0,0.0) in D, (5.0,0.0) in E and (20.0,2.0) in F. β = 1.0 × 104, λini = λg = 10.0, τ = 300, and σr = σθ = 0.05 in D–F.
Fig 3.
A, B. The number of primordia before the first arrest (arrowheads in Fig 2C and 2E) is depicted by colors in the legend. The red region indicates a non-whorled pattern. For simplicity, we set PMP = 0 (Eq 4) so that primordia could not move against the potential gradient Ug, k. λini = λg = 10.0, σr = σθ = 0.05. α = 0.0 (A) and α = 2.0 (B). The four panels between A and B are representative examples of each merosity where the arrowhead indicates the third primordium. C. Phase diagram of the first-whorl merosity according to α and R0/Vτ at (white line in A). The color code is the same as that in A and B. The region of dimerous arrangement (green) increases as α increases, because the previous primordium becomes the most dominant inhibitor so that the new primordium initiates just opposite to the previous one and its growth is arrested by the second previous one.
Fig 4.
Reconstructing pentamerous floral development.
A. Flower of Silene coeli-rosa (Caryophyllaceae). B. Reproduction of the S. coeli-rosa floral meristem traced from an SEM image by Lyndon [34]; the colors were modified. Numbers indicate the initiation order. K (sepals), C (petals), St (stamens), AB (axillary bud). C. Average position of the S. coeli-rosa floral primordia reconstructed from the divergence angle and plastochron ratio (E) measured by Lyndon (Table 1 in [34]). The number of measured apices is 9 for sepals, 5 for petals, 7 for stamens, and 2 for carpels. The positions of sepals and petals are depicted in large squares, and those of stamens and carpels are depicted in small squares. D. Spatial pattern of the model simulation. The first ten primordia are shown by large circles, and the subsequent ten primordia are shown by small circles. τ = 600, R0 = 30.0, α = 3.0, σr = 0.05, σθ = 5.0, λini = λg = 20.0, PMP = 0. E. Divergence angle (top panel) and plastochron ratio (middle) between two succeeding primordia, and the distance from the center of the apex (bottom panel) in S. coeli-rosa (blue squares) and in the model simulation (red circles). The order of petal initiation was estimated from that of the adjacent stamens (St6-St10 in B) following the experimental report [34]. The measurements agree with the model until the ninth primordium (open arrowhead). Error bars for the divergence angle and plastochron ratio of S. coeli-rosa denote the standard errors. Because the absolute values of the S. coeli-rosa primordia radii were not published, the distance from the center is normalized by the radius of the first sepal. The values of the parameters are the same as those in D. The green line (D and E bottom panel) indicates the meristem boundary in the simulation.
Fig 5.
The potential landscape captures tetramerous whorl formation.
A–C. Color-coded potential landscape (upper panel; legend) and the section (bottom panel) at the angle where Uini takes the global minimum so that the fifth primordium arises (white dashed line in upper panel; Eq 6). The green line shows the meristem edge with a diameter of R0. The direction of the potential at the position where the fifth primordium arises, denoted by the red circle, is inward in B but outward in A and C (bottom panel). R0 = 5.0 (A), 15.0 (B), 45.0 (C). D. Radial positions that take the local minima (rmin, blue circles) and maxima (rmax, red squares) of potential Ug,5 (Eq 6). Between R0 = rmin and rmax, indicated by the two arrowheads, the potential at the meristem edge decreases inward as in B. Black diamonds correspond to the initiating position of the fifth primordium of A–C. Vτ = 6.0, σθ = 0.0, λini = λg = 10.0, and PMP = 0 in A–D. E. Superposition of the analytical result onto the numerical results (Fig 3A). Solid lines show the crossovers rmin = R0 and rmax = R0, respectively (arrowheads in D). λini = λg = 10.0 and PMP = 0. σr = σθ = 0.05 for numerical result, σθ = 0.0 for analytical result.
Fig 6.
The potential landscape captures pentamerous whorl formation.
A. The angular position of the third primordium as a function of α. B–C. Color-coded growth-potential landscape (top) and the section (bottom) at the angle where the fifth (B) and the sixth (C) primordia arise (white dashed line in the top panel). α = 2.0, Vτ = 6.0, R0 = 20.0, σθ = 0.0, λini = λg = 10.0, and PMP = 0 in A–C.
Fig 7.
Effects of λini, λg, and α on merosities.
Superposition of the analytical result (the solid lines are identical to Fig 5E: λini = λg = 10.0, α = 0.0, PMP = 0) and the numerical result (σr = σθ = 0.05; the following parameters are different from the solid line: A. λini = 5.0, B. λini = 20.0, C. α = 2.0 D. α = 2.0, λini = 20.0, E. λg = 5.0, F. λg = 20.0). The colors follow Fig 3. λg and α affect the boundary lines between each whorl as well as that between non-whorls and tetramerous whorls (C, E and F), whereas λini hardly affects at α = 0 (A and B). At α ≠ 0, α and λini synergistically affect the phase boundaries (C and D).