Skip to main content
  • Submit
  • About
    • Editorial Board
    • PNAS Staff
    • FAQ
    • Accessibility Statement
    • Rights and Permissions
    • Site Map
  • Contact
  • Journal Club
  • Subscribe
    • Subscription Rates
    • Subscriptions FAQ
    • Open Access
    • Recommend PNAS to Your Librarian
  • Log in
  • My Cart

Main menu

  • Home
  • Articles
    • Current
    • Special Feature Articles - Most Recent
    • Special Features
    • Colloquia
    • Collected Articles
    • PNAS Classics
    • List of Issues
  • Front Matter
  • News
    • For the Press
    • This Week In PNAS
    • PNAS in the News
  • Podcasts
  • Authors
    • Information for Authors
    • Editorial and Journal Policies
    • Submission Procedures
    • Fees and Licenses
  • Submit
  • About
    • Editorial Board
    • PNAS Staff
    • FAQ
    • Accessibility Statement
    • Rights and Permissions
    • Site Map
  • Contact
  • Journal Club
  • Subscribe
    • Subscription Rates
    • Subscriptions FAQ
    • Open Access
    • Recommend PNAS to Your Librarian

User menu

  • Log in
  • My Cart

Search

  • Advanced search
Home
Home

Advanced Search

  • Home
  • Articles
    • Current
    • Special Feature Articles - Most Recent
    • Special Features
    • Colloquia
    • Collected Articles
    • PNAS Classics
    • List of Issues
  • Front Matter
  • News
    • For the Press
    • This Week In PNAS
    • PNAS in the News
  • Podcasts
  • Authors
    • Information for Authors
    • Editorial and Journal Policies
    • Submission Procedures
    • Fees and Licenses

New Research In

Physical Sciences

Featured Portals

  • Physics
  • Chemistry
  • Sustainability Science

Articles by Topic

  • Applied Mathematics
  • Applied Physical Sciences
  • Astronomy
  • Computer Sciences
  • Earth, Atmospheric, and Planetary Sciences
  • Engineering
  • Environmental Sciences
  • Mathematics
  • Statistics

Social Sciences

Featured Portals

  • Anthropology
  • Sustainability Science

Articles by Topic

  • Economic Sciences
  • Environmental Sciences
  • Political Sciences
  • Psychological and Cognitive Sciences
  • Social Sciences

Biological Sciences

Featured Portals

  • Sustainability Science

Articles by Topic

  • Agricultural Sciences
  • Anthropology
  • Applied Biological Sciences
  • Biochemistry
  • Biophysics and Computational Biology
  • Cell Biology
  • Developmental Biology
  • Ecology
  • Environmental Sciences
  • Evolution
  • Genetics
  • Immunology and Inflammation
  • Medical Sciences
  • Microbiology
  • Neuroscience
  • Pharmacology
  • Physiology
  • Plant Biology
  • Population Biology
  • Psychological and Cognitive Sciences
  • Sustainability Science
  • Systems Biology
Research Article

A plausible model of phyllotaxis

Richard S. Smith, Soazig Guyomarc'h, Therese Mandel, Didier Reinhardt, Cris Kuhlemeier, and Przemyslaw Prusinkiewicz
PNAS January 31, 2006 103 (5) 1301-1306; first published January 23, 2006; https://doi.org/10.1073/pnas.0510457103
Richard S. Smith
*Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, AB, Canada T2N 1N4; and‡Institute of Plant Sciences, University of Berne, CH-3013 Berne, Switzerland
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
Soazig Guyomarc'h
*Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, AB, Canada T2N 1N4; and‡Institute of Plant Sciences, University of Berne, CH-3013 Berne, Switzerland
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
Therese Mandel
*Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, AB, Canada T2N 1N4; and‡Institute of Plant Sciences, University of Berne, CH-3013 Berne, Switzerland
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
Didier Reinhardt
*Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, AB, Canada T2N 1N4; and‡Institute of Plant Sciences, University of Berne, CH-3013 Berne, Switzerland
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
Cris Kuhlemeier
*Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, AB, Canada T2N 1N4; and‡Institute of Plant Sciences, University of Berne, CH-3013 Berne, Switzerland
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
Przemyslaw Prusinkiewicz
*Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, AB, Canada T2N 1N4; and‡Institute of Plant Sciences, University of Berne, CH-3013 Berne, Switzerland
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
  1. Communicated by Enrico Sandro Coen, John Innes Centre, Norwich, United Kingdom, December 6, 2005 (received for review October 22, 2005)

  • Article
  • Figures & SI
  • Info & Metrics
  • PDF
Loading

Article Figures & SI

Figures

  • Fig. 1.
    • Download figure
    • Open in new tab
    • Download powerpoint
    Fig. 1.

    Conceptual model of the regulation of phyllotaxis by polar auxin fluxes in the shoot meristem. Adapted from ref. 10. (A) PIN1 orientation directs auxin fluxes (arrows) in the L1 layer, leading to accumulation of auxin (red color) at the initiation site (I1) in the peripheral zone. This accumulation eventually results in organ induction. (B) Later, basipetal PIN1 polarization inside the bulging primordium (P1) drains auxin into inner layers, depleting the neighboring L1 cells. As a consequence, another auxin maximum is created in the peripheral zone at position I1 removed from primordia P1 and P2.

  • Fig. 2.
    • Download figure
    • Open in new tab
    • Download powerpoint
    Fig. 2.

    DR5::GFP expression in the shoot apex. (A) Longitudinal section of a DR5::GFP-expressing wild-type inflorescence meristem. The arrowhead indicates local overexpression of DR5::GFP consistent with an initial. (B) Longitudinal section of a DR5::GFP-expressing wild-type inflorescence meristem treated with 25 μM sirtinol during 48 h. (C, D, F, and G) Transverse confocal pictures, taken with comparable settings. (C) DR5::GFP expression pattern in a pin1-7 inflorescence meristem. (D) DR5::GFP expression pattern in a wildtype inflorescence meristem treated with 20 μM NPA during 24 h. (E) Top view of a wild-type inflorescence meristem, visualized with a binocular microscope. (F) Transverse confocal picture showing the DR5::GFP expression pattern in the same meristem as shown in the frame indicated in E. Spots of GFP signal are observed in the peripheral zone of the meristem where no bulge is visible (compare to E). (G) DR5::GFP expression pattern in a wild-type inflorescence meristem treated with 25 μM sirtinol during 48 h. (H) Longitudinal view of the two first, 3-day-old, leaf primordia of a DR5::GFP-expressing seedling. Note the strong signals at the tips of the primordia (arrowheads), and DR5::GFP expression in the future midveins (arrows), which connect with the stem vasculature (asterisk). P indicates the DR5::GFP expression in the next primordium. (Scale bars: 25 μm.)

  • Fig. 3.
    • Download figure
    • Open in new tab
    • Download powerpoint
    Fig. 3.

    Pattern generation by the transport-based model. (A–D) Pattern emergence in a sequence of 50 cells with wraparound boundary conditions (the leftmost and the rightmost cell are considered neighbors). Taller bars (brighter green) indicate higher IAA concentration. Simulation steps 30, 60, 70, and 80 are shown. A small amount of noise present in the initial distribution is required to break symmetry. (E and F) Pattern dependence on model parameters. Model parameters affect how many peaks a given number of cells will create. Higher values of the transport coefficient result in more peaks. If the transport coefficient is too low, no peaks will form at all. Transport coefficient: A-D, 4.0; E, 3.0; F, 10.0. (G) Pattern formed in a simulated cellular structure. PIN1 is depicted in red.

  • Fig. 4.
    • Download figure
    • Open in new tab
    • Download powerpoint
    Fig. 4.

    Simulated shoot apical meristems. (A–D) The arrangement of primordia into phyllotactic patterns: distichous (A), decussate (B), tricussate (C), and spiral (D). (E–G) Simulated pin1 mutant (E), primordium formed in the pin1 mutant after localized application of auxin in the peripheral zone (F), and primordium ring formed in the pin1 mutant after localized application of auxin at the tip of the apex (G). Arrows indicate the site of auxin application. (H and I) Simulated results of cell ablation: control apex (H) and the apex in which five cells shown in black have been removed (I). Note the shift in the position of the primordium initial (cells with high auxin concentration, shown in bright green) near the ablation site.

  • Fig. 5.
    • Download figure
    • Open in new tab
    • Download powerpoint
    Fig. 5.

    Comparison of the divergence angles: angles measured in Arabidopsis with standard error bars (blue), and angles generated by the spiral phyllotaxis model (red).

Data supplements

  • Smith et al. 10.1073/pnas.0510457103.

    Supporting Information

    Files in this Data Supplement:

    Supporting Figure 6
    Supporting Text
    Supporting Figure 7
    Supporting Figure 8
    Supporting Movie 1
    Supporting Movie 2
    Supporting Movie 3
    Supporting Movie 4
    Supporting Table 1
    Supporting Table 2
    Supporting Table 3




    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.





    Supporting Movie 1

    Movie 1. Distichous phyllotaxis simulation.





    Supporting Movie 2

    Movie 2. Decussate phyllotaxis simulation.





    Supporting Movie 3

    Movie 3. Tricussate phyllotaxis simulation.





    Supporting Movie 4

    Movie 4. Fibonacci spiral phyllotaxis simulation.





    Table 1. Model parameters for phyllotaxis simulations

    Parameter

    Eq.

    Distichous

    Decussate

    Tricussate

    Spiral

    IAA production coefficient in the proximal zone

    ρIAA(proximal)

    5

    1.000

    1.000

    3.000

    3.000

    IAA production coefficient in the peripheral zone

    ρIAA(peripheral)

    5

    5.000

    6.000

    7.500

    6.000

    Coefficient of additional IAA production in primordia

    ρIAA(primordium)

    6

    1.000

    0.000

    0.000

    3.500

    Coefficient controlling saturation of IAA production

    κIAA

    5, 6

    1.000

    1.000

    1.000

    1.000

    IAA decay coefficient

    μIAA

    5

    0.100

    0.100

    0.100

    0.100

    IAA diffusion coefficient per unit cell wall length

    D

     

    4.000

    4.000

    4.000

    4.000

    IAA transport coefficient

    T

    3

    40.000

    22.500

    22.000

    22.500

    Coefficient controlling IAA transport saturation

    κT

    3

    1.000

    1.000

    1.000

    1.000

    IAA threshold for primordium differentiation

    Th

     

    7.500

    8.000

    7.000

    7.500

    Maximum IAA concentration

    [IAA]max

    7

    20.000

    20.000

    20.000

    15.000

    Base production of PIN

    ρPIN0

    1

    0.500

    0.500

    0.600

    0.500

    Coefficient of PIN production depending on IAA concentration

    ρPIN

    1

    0.025

    0.025

    0.010

    0.025

    Coefficient controlling saturation of PIN production

    κPIN

    1

    1.000

    1.000

    1.000

    1.000

    PIN decay coefficient

    μPIN

    1

    0.100

    0.350

    0.350

    0.350

    Exponentiation base for calculating PIN relocation

    b

    2

    3.000

    3.000

    3.000

    3.000

    Top border of peripheral zone*

     

     

    2.500

    2.800

    4.100

    2.900

    Bottom border of peripheral zone†

     

     

    5.000

    8.400

    12.300

    8.700

    Parameters without associated symbols are not further described in the article. The simulations use different functions to control the active ring position and size during startup. All parameters for rendering, surface and primordium shape, and surface growth are the same for all simulations. Distichous parameters are used for Movie 1 and Fig. 4A. Decussate parameters are used for Movie 2 and Fig. 4B. Tricussate parameters are used for Movie 3 and Fig. 4C. Spiral parameters are used for Movie 4, Fig. 4 D, H, and I (see also Fig. 5), and Table 2. Spiral parameters with PIN production and decay set to 0 are used for Fig. 4 E and F.
    *Steady-state distance between the apex tip and the top of the peripheral zone, measured along the generating curve.
    †Steady-state distance between the apex tip and the bottom of the peripheral zone, measured along the generating curve.





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

    Primordium number

    Divergence angle, °

    Time, simulation
    steps × 100

    1

    0.000

    2.78

    2

    162.740

    3.00

    3

    96.272

    6.81

    4

    155.880

    7.20

    5

    123.397

    10.48

    6

    143.205

    11.05

    7

    139.269

    14.03

    8

    128.599

    14.87

    9

    142.127

    16.96

    10

    124.750

    18.48

    11

    144.179

    19.79

    12

    131.910

    21.46

    13

    125.384

    22.91

    14

    141.180

    24.34

    15

    132.346

    26.35

    16

    131.631

    27.37

    17

    124.991

    29.53

    18

    137.307

    30.88

    19

    135.372

    32.73

    20

    124.094

    34.03

    21

    122.975

    35.64

    22

    144.799

    37.08

    23

    114.642

    38.72

    24

    128.862

    40.15

    25

    133.097

    42.00

    26

    134.966

    43.74

    27

    142.972

    45.61

    28

    150.016

    47.05

    29

    130.865

    49.26

    30

    123.537

    50.10

    31

    132.814

    52.04

    32

    126.695

    53.46

    33

    136.362

    55.01

    34

    130.081

    57.07

    35

    134.563

    58.35

    36

    130.904

    60.03

    37

    135.225

    61.30

    38

    125.441

    63.02

    39

    153.168

    64.63

    40

    126.086

    66.63





    Table 3. Divergence angles in degrees for sensitivity analysis simulations

    Primordium number

    Simulation

    11

    12

    13

    14

    15

    16

    17

    18

    19

    20

    SA1

    145.093

    143.843

    139.470

    133.792

    143.319

    134.723

    130.035

    125.708

    137.444

    144.262

    SA2

    137.329

    128.618

    138.899

    122.192

    143.398

    139.259

    122.611

    143.455

    126.868

    135.962

    SA3

    127.730

    145.353

    125.170

    136.259

    130.892

    126.798

    135.469

    124.181

    148.881

    122.814

    SA4

    144.026

    131.154

    134.904

    119.808

    145.070

    137.162

    124.992

    126.194

    128.128

    133.705





    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 ρIAA(proximal) from 3.0 to 2.5 and additional IAA production within primordia ρIAA(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 κIAA to zero. Spiral phyllotaxis was maintained after reducing IAA proximal zone production ρIAA(proximal) from 3.0 to 1.0, IAA production in the peripheral zone ρIAA(peripheral) from 3.0 to 0.9, and additional IAA production within primordia ρIAA(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 κIAA 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.

    1. Benková, E., Michniewicz, M., Sauer, M., Teichmann, T., Seifertova, D., Jurgens, G. & Friml, J. (2003) Cell115, 591–602.

    2. Ulmasov, T., Murfett, J., Hagen, G. & Guilfoyle, T. J. (1997) Plant Cell9, 1963–1971.

    3. Smith, C., Prusinkiewicz, P. & Samavati, F. (2003) Lecture Notes in Computer Science (Springer, Berlin) Vol. 3062, pp. 313–327.

    4. Kramer, E. M. (2002) J. Theor. Biol.216, 147–158.

    5. Nakielski, J. (2000) in Pattern Formation in Biology, Vision, and Dynamics, eds. Carbone, A., Gromov, M. & Prusinkiewicz, P. (World Scientific, Singapore), pp. 252–267.

    6. Hejnowicz, Z., Nakielski, J. & Hejnowicz, K. (1984) Acta Soc. Bot. Pol.53, 301–316.

    7. Mündermann, L., Erasmus, Y., Lane, B., Coen, E. & Prusinkiewicz, P. (2005) Plant Physiol.139, 960–968.

    8. Silk, W. K. & Erickson, R. O. (1979) J. Theor. Biol.76, 481–501.

    9. Richards, O. W. & Kavanagh, A. J. (1943) Amer. Nat.77, 385–399.

    10. Kwiatkowska, D. & Dumais, J. (2003) J. Exp. Bot.54, 1585–1595.

    11. Reddy, G. V., Heisler, M. G., Ehrhardt, D. W. & Meyerowitz, E. M. (2004) Development(Cambridge, U.K.)131, 4225–4237.

PreviousNext
Back to top
Article Alerts
Email Article

Thank you for your interest in spreading the word on PNAS.

NOTE: We only request your email address so that the person you are recommending the page to knows that you wanted them to see it, and that it is not junk mail. We do not capture any email address.

Enter multiple addresses on separate lines or separate them with commas.
A plausible model of phyllotaxis
(Your Name) has sent you a message from PNAS
(Your Name) thought you would like to see the PNAS web site.
CAPTCHA
This question is for testing whether or not you are a human visitor and to prevent automated spam submissions.
Citation Tools
A plausible model of phyllotaxis
Richard S. Smith, Soazig Guyomarc'h, Therese Mandel, Didier Reinhardt, Cris Kuhlemeier, Przemyslaw Prusinkiewicz
Proceedings of the National Academy of Sciences Jan 2006, 103 (5) 1301-1306; DOI: 10.1073/pnas.0510457103

Citation Manager Formats

  • BibTeX
  • Bookends
  • EasyBib
  • EndNote (tagged)
  • EndNote 8 (xml)
  • Medlars
  • Mendeley
  • Papers
  • RefWorks Tagged
  • Ref Manager
  • RIS
  • Zotero
Request Permissions
Share
A plausible model of phyllotaxis
Richard S. Smith, Soazig Guyomarc'h, Therese Mandel, Didier Reinhardt, Cris Kuhlemeier, Przemyslaw Prusinkiewicz
Proceedings of the National Academy of Sciences Jan 2006, 103 (5) 1301-1306; DOI: 10.1073/pnas.0510457103
Digg logo Reddit logo Twitter logo Facebook logo Google logo Mendeley logo
  • Tweet Widget
  • Facebook Like
  • Mendeley logo Mendeley
Proceedings of the National Academy of Sciences of the United States of America: 103 (5)
Table of Contents

Submit

Sign up for Article Alerts

Jump to section

  • Article
    • Abstract
    • Experimental Results
    • Simulation Model
    • Conclusions
    • Materials and Methods
    • Supplementary Material
    • Acknowledgments
    • Footnotes
    • References
  • Figures & SI
  • Info & Metrics
  • PDF

You May Also be Interested in

Abstract depiction of a guitar and musical note
Science & Culture: At the nexus of music and medicine, some see disease treatments
Although the evidence is still limited, a growing body of research suggests music may have beneficial effects for diseases such as Parkinson’s.
Image credit: Shutterstock/agsandrew.
Scientist looking at an electronic tablet
Opinion: Standardizing gene product nomenclature—a call to action
Biomedical communities and journals need to standardize nomenclature of gene products to enhance accuracy in scientific and public communication.
Image credit: Shutterstock/greenbutterfly.
One red and one yellow modeled protein structures
Journal Club: Study reveals evolutionary origins of fold-switching protein
Shapeshifting designs could have wide-ranging pharmaceutical and biomedical applications in coming years.
Image credit: Acacia Dishman/Medical College of Wisconsin.
White and blue bird
Hazards of ozone pollution to birds
Amanda Rodewald, Ivan Rudik, and Catherine Kling talk about the hazards of ozone pollution to birds.
Listen
Past PodcastsSubscribe
Goats standing in a pin
Transplantation of sperm-producing stem cells
CRISPR-Cas9 gene editing can improve the effectiveness of spermatogonial stem cell transplantation in mice and livestock, a study finds.
Image credit: Jon M. Oatley.

Similar Articles

Site Logo
Powered by HighWire
  • Submit Manuscript
  • Twitter
  • Facebook
  • RSS Feeds
  • Email Alerts

Articles

  • Current Issue
  • Latest Articles
  • Archive

PNAS Portals

  • Anthropology
  • Chemistry
  • Classics
  • Front Matter
  • Physics
  • Sustainability Science
  • Teaching Resources

Information

  • Authors
  • Editorial Board
  • Reviewers
  • Librarians
  • Press
  • Site Map
  • PNAS Updates

Feedback    Privacy/Legal

Copyright © 2021 National Academy of Sciences. Online ISSN 1091-6490