A plausible model of phyllotaxis

Supporting Information

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Fig. 6. Series of transverse optical sections through DR5::GFP-expressing inflorescence meristems, followed by the overlay of these sections (E, J, O, and T). (A–E) Wild-type inflorescence meristem. (F–J) Wild-type inflorescence meristem treated with 25 μM sirtinol during 48 h. (K–O) Wild-type inflorescence meristem treated with 20 μM naphthylphthalamic acid (NPA) during 24 h. (P–T) Inflorescence meristem of a pin1-7 mutant. (Scale bars: 25 μm.)

Fig. 7. (A) Geometric model of the shoot apical meristem. The reference surface is obtained by revolving a generating curve around the surface axis. Primordia (green) are modeled as smaller surfaces of revolution, protruding from the reference surface. Growth of the reference surface is modeled as a flow and elongation of surface elements (red arrows). This flow also defines the gradual displacement of the primordia over time, as their radius r and height h gradually increase (r₁ > r₂ and h₁ > h₂). Cells (dark polygons) grow as a result of the expansion of the apex surface and divide upon reaching the maximum size. (B) Details of the model of cell division. (i) The mother cell before division. (ii) The tentative dividing wall is the shortest wall passing through the centroid of the mother cell. (iii) The endpoints of the tentative wall are displaced from the vertices of the mother cell. (iv) The form of daughter cells is adjusted by shortening the dividing wall.

Fig. 8. Hypothetical feedback loops in cell polarization and auxin transport. A sample cell i and two of its neighbors, cells j and k, are shown. PIN1 molecules in each cell are distributed between cell membranes according to auxin concentration in the neighboring cells (control information shown in red). The resulting auxin fluxes Ф modify auxin concentrations in each cell. These concentrations affect, in turn, both the concentration of PIN1 molecules in a given cell and distribution of PIN1 molecules in the neighboring cells. The model includes two types of feedback loops: local to a cell and spatially distributed, involving neighboring cells.

Movie 1. Distichous phyllotaxis simulation.

Movie 2. Decussate phyllotaxis simulation.

Movie 3. Tricussate phyllotaxis simulation.

Movie 4. Fibonacci spiral phyllotaxis simulation.

Table 1. Model parameters for phyllotaxis simulations

Table 2. Divergence angles and primordium differentiation times for spiral phyllotaxis simulation

Table 3. Divergence angles in degrees for sensitivity analysis simulations

Supporting Text

Supporting Materials and Methods

Lines. All Arabidopsis thaliana lines were in the Columbia background. The pin1 mutant allele used was pin1-7. DR5::GFP refers to the DR5rev::GFP line (1), in which the DR5 promoter consists of nine repeats of the DR5 element (2), fused in inverse orientation to the minimal CaMV 35S promoter and the coding sequence of an endoplasmic-reticulum-targeted eGFP protein. This construct was kindly provided by J. Friml (Universität Tübingen, Tübingen, Germany) and crossed into the pin1-7 background.

Treatments. N-1-naphthylphthalamic acid (NPA, 100 mM) and sirtinol (10 mM) stock solutions in dimethylsulphoxide (DMSO) were dissolved in half-strength MS medium (Serva, catalog no. 47515) to a final concentration of 20 μM and 25 μM, respectively. As a control, plates containing the same volume of DMSO dissolved in the medium were used. Immediately after dissection, the inflorescence meristems were grown on these plates or on normal one-half MS plates for 4, 24, or 48 h and then observed for DR5::GFP expression.

Confocal Microscopy. DR5::GFP expression patterns were studied by using a Leica upright confocal laser-scanning microscope (DMRXE7), equipped with a long-working-distance water immersion objective (×L63/0.90 U-V-I). The selected wavelengths were 500–550 nm for GFP and 620–690 nm for chlorophyll (used to check the specificity of the GFP signal).

Divergence Angle Measurements. Divergence angles were measured from sequential photographs taken at 2-day intervals beginning with seedlings 10 days after germination. A minimum of 22 plants was measured for each angle.

Computer Programming. Simulation models were implemented by using the VV programming environment (3), which extends the C++ language with support for specifying and visualizing dynamically changing cellular structures. Movies were assembled from sequences of frames output by VV with the TMPGENC video editing software. All simulations were performed on a 3 GHz Pentium 4 PC with an ATI Radeon 9000 graphics card.

Geometric Model of a Growing Meristem

Leaf initiation takes place in the peripheral zone of a growing shoot apical meristem and involves the patterning of cells that are rapidly dividing. According to the molecular data, the L1 layer plays a key role in this process. For the purposes of simulation, we treat this layer as a curved surface of negligible thickness, which allows us to simplify the problem of morphogenesis from three dimensions to two (4). The geometric model has four components: a model of the basic shape of the apex, a model of apex growth, a model of primordium outgrowth, and a model of cell shape and division.

The shape of the shoot apex is approximated as a rotationally symmetric reference surface (5–7) with superimposed outward-growing primordia (Fig. 7A). The reference surface is an idealization of the shape of the apex in the absence of primordia.

Growth of the reference surface is simulated by moving points embedded in it basipetally while maintaining the overall surface shape. This process is characterized by a relative elementary rate of growth function RERG(x), which defines the rate of elongation of infinitesimal segments of the generating curve as a function of their distance from the apex tip (6, 8, 9). The velocity with which surface points move away from the tip is obtained by integrating the RERG function along the generating curve. Following experimental data (10, 11), the RERG distribution is chosen such that the growth in the central zone is slower than in the peripheral zone.

Primordia are modeled as growing, rotationally symmetric bulges on the reference surface (Fig. 7A). A primordium center is placed at the average of the centroids of two adjacent cells that have an IAA concentration above the primordium differentiation threshold, Th. The radius r and height h of the primordium increase with the primordium age. These parameters act as scaling factors for a sigmoidal curve that defines the longitudinal cross section of the primordium.

Cells are modeled as polygons, following the method of Nakielski (5). The position of cell vertices changes over time as a result of the growth of the reference surface and gradual protrusion of primordia (Fig. 7A). Cell division occurs when the parent cell size (polygon area) reaches a threshold value. By default, the dividing wall is the shortest wall that passes through the centroid of the cell polygon. This position may be adjusted to avoid four-way junctions, which are unusual (11). To produce more realistic cell shapes, the vertices of the dividing wall are slightly moved toward each other (Fig. 7B). As observed by Nakielski (5), the dynamics of cell division and the resulting cellular patterns are similar to those observed in the Arabidopsis meristem (10, 11). We thus consider them an adequate structural support for modeling the auxin fluxes and PIN1 localization during phyllotactic pattern formation.

Sensitivity Analysis

To provide insight into the form and role of parameters in the model equations, we performed a preliminary sensitivity analysis. Starting with the parameter values for the spiral phyllotaxis simulation (column Spiral in Table 1), selected equations or parameters of the model were changed, and simulations were run again to determine whether spiral phyllotaxis could be maintained. Invariably, additional parameters needed to be adjusted, but an attempt was made to minimize their number. We focused on the analysis of pattern maintenance rather than pattern initiation. Consequently, the simulations were started by placing 10 initial primordia in a spiral pattern. Four simulated experiments were considered, and the divergence angles of primordia 11-20 observed (Table 3).

Fixed PIN1 Concentration. In this simulation, the equation for PIN1 production and decay (Eq. 1) was removed from the model, and PIN1 concentration was fixed at 0.94, the approximate value for nonprimordium cells in the original spiral model. Spiral phyllotaxis was maintained after decreasing IAA proximal zone production ρ_IAA(proximal) from 3.0 to 2.5 and additional IAA production within primordia ρ_IAA(primordium) from 3.5 to 2.5. The resulting sequence of divergence angles is listed in row SA1 of Table 3.

After fixing PIN1 concentration, the new parameter values that resulted in pattern maintenance were not difficult to find. On this basis, we concluded that the exact form of Eq. 1 is not critical to phyllotactic pattern formation.

Removal of IAA Production Saturation Term. According to Eqs. 5 and 6, high IAA concentration saturates IAA production. We investigated the case when no saturation was present by setting the IAA production saturation coefficient κ_IAA to zero. Spiral phyllotaxis was maintained after reducing IAA proximal zone production ρ_IAA(proximal) from 3.0 to 1.0, IAA production in the peripheral zone ρ_IAA(peripheral) from 3.0 to 0.9, and additional IAA production within primordia ρ_IAA(primordium) from 3.5 to 1.0 (simulation SA2 in Table 3).

As in the case of the simulation assuming fixed PIN1 concentration, it was not difficult to find parameter values that lead to the maintenance of spiral phyllotaxis. However, with κ_IAA set to zero, the model was no longer able to reproduce the pin1 mutant phenotype. The IAA concentration was building up, leading to continuous formation of primordia that were completely filling the peripheral zone.

Reduction of Transport Exponent. The numerator in the IAA transport equation (Eq. 3) specifies that the flux of IAA out of a source cell i is proportional to the square of the IAA concentration in this cell. Similarly, the denominator specifies that the flux of IAA to a destination cell j depends on the square of the IAA concentration in that cell.

To investigate whether the model critically depends on these quadratic relations, we decreased the exponent value from 2.0 to 1.5. The spiral phyllotaxis was maintained after reducing the transport coefficient T' from 22.5 to 19.0, the primordium differentiation threshold Th from 7.5 to 7.0, and the peripheral zone size from 2.9 to 2.57 (simulation SA3 in Table 3) . However, the model was very sensitive to parameter values, possibly due to the reduced cell count in the smaller peripheral zone.

Increase in PIN1 Relocation Exponent. The PIN1 polarization equation (Eq. 2) gives an exponential preference to the neighbor cells with the highest IAA concentration. Experiments with other formulas, in which the polarization depended on the neighbor IAA concentration according to a linear or power function, did not yield spiral phyllotactic patterns.

To investigate how critically the model depends on the value of the exponentiation base b (Eq. 2), we increased it from 3.0 to 4.0. Spiral phyllotaxis was maintained after increasing diffusion coefficient D from 4.0 to 4.5 and decreasing transport coefficient T from 22.5 to 22 (simulation SA4 in Table 3), but the simulation was very sensitive to parameter values. Further experiments have shown that decussate and tricussate patterns could be obtained for b values as high as 5.0, but spiral simulations were most stable at values close to 3.0. Reducing the exponentiation base b below 2.5 did not yield spiral phyllotactic patterns.

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