- Open Access
Phyllotaxis in a Keller-Segel model
Phys. Rev. E 113, 034407 – Published 27 March, 2026
DOI: https://doi.org/10.1103/9yk5-b4q2
Abstract
Plants often exhibit regular arrangements of seeds or leaves, known as phyllotaxis, and can famously exhibit Fibonacci spirals, as in sunflower heads or on pine cones. While there are many models which can reproduce a spiral formation, the actual mechanism of arrangement remains unclear. Here, we test a more general class of model, based on the Keller-Segel model, in which a diffusing chemical signal produces its own transport signal and can form local instabilities. By modeling the plant as a disk with a signal source at the center and radial growth, we show that we can reproduce both Fibonacci spirals and alternating patterns, depending on a balance between growth and source. Finally, using linear stability analysis, we show how the pattern arises due to a balance between the growth of instabilities and the growth of the plant. Overall, this work demonstrates how the Keller-Segel model can reproduce a wide range of observed patterns, and may act as a phenomenological description of a number of processes in plants.
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References (26)
- L. Levitov, Fibonacci numbers in botany and physics: Phyllotaxis, JETP Lett. 54, 546 (1991).
- I. Adler, D. Barabe, and R. V. Jean, A history of the study of phyllotaxis, Ann. Bot. 80, 231 (1997).
- M. F. Pennybacker, P. D. Shipman, and A. C. Newell, Phyllotaxis: Some progress, but a story far from over, Physica D 306, 48 (2015).
- W. Hofmeister, Allgemeine Morphologie der Gewächse (Engelmann, Leipzig, 1868), Vol. 1.
- H. Airy, On leaf-arrangement, Proc. R. Soc. London 21, 176 (1873).
- G. Van Iterson, Mathematische und mikroskopisch-anatomische Studien über Blattstellungen nebst Betrachtungen über den Schalenbau der Miliolinen (Fischer, Jena, 1907).
- S. Douady and Y. Couder, Phyllotaxis as a physical self-organized growth process, Phys. Rev. Lett. 68, 2098 (1992).
- L. Levitov, Energetic approach to phyllotaxis, Europhys. Lett. 14, 533 (1991).
- C. Nisoli, N. M. Gabor, P. E. Lammert, J. Maynard, and V. H. Crespi, Annealing a magnetic cactus into phyllotaxis, Phys. Rev. E 81, 046107 (2010).
- J. Dumais and C. R. Steele, New evidence for the role of mechanical forces in the shoot apical meristem, J. Plant Growth Regul. 19, 7 (2000).
- P. Shipman and A. Newell, Polygonal planforms and phyllotaxis on plants, J. Theor. Biol. 236, 154 (2005).
- Y. Zhao, Auxin biosynthesis and its role in plant development, Annu. Rev. Plant Biol. 61, 49 (2010).
- T. Vernoux, F. Besnard, and J. Traas, Auxin at the shoot apical meristem, Cold Spring Harbor Perspect. Biol. 2, a001487 (2010).
- V. Pinon, K. Prasad, S. P. Grigg, G. F. Sanchez-Perez, and B. Scheres, Local auxin biosynthesis regulation by plethora transcription factors controls phyllotaxis in Arabidopsis, Proc. Natl. Acad. Sci. USA 110, 1107 (2013).
- D. Reinhardt, E.-R. Pesce, P. Stieger, T. Mandel, K. Baltensperger, M. Bennett, J. Traas, J. Friml, and C. Kuhlemeier, Regulation of phyllotaxis by polar auxin transport, Nature (London) 426, 255 (2003).
- R. S. Smith, S. Guyomarc'h, T. Mandel, D. Reinhardt, C. Kuhlemeier, and P. Prusinkiewicz, A plausible model of phyllotaxis, Proc. Natl. Acad. Sci. USA 103, 1301 (2006).
- A. C. Newell, P. D. Shipman, and Z. Sun, Phyllotaxis: Cooperation and competition between mechanical and biochemical processes, J. Theor. Biol. 251, 421 (2008).
- H. Jönsson, M. G. Heisler, B. E. Shapiro, E. M. Meyerowitz, and E. Mjolsness, An auxin-driven polarized transport model for phyllotaxis, Proc. Natl. Acad. Sci. USA 103, 1633 (2006).
- M. G. Heisler, O. Hamant, P. Krupinski, M. Uyttewaal, C. Ohno, H. Jönsson, J. Traas, and E. M. Meyerowitz, Alignment between PIN 1 polarity and microtubule orientation in the shoot apical meristem reveals a tight coupling between morphogenesis and auxin transport, PLoS Biol. 8, e1000516 (2010).
- M. Pennybacker and A. C. Newell, Phyllotaxis, pushed pattern-forming fronts, and optimal packing, Phys. Rev. Lett. 110, 248104 (2013).
- D. T. Lamport, L. Tan, M. Held, and M. J. Kieliszewski, Phyllotaxis turns over a new leaf—a new hypothesis, Int. J. Mol. Sci. 21, 1145 (2020).
- E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol. 30, 225 (1971).
- M. A. Herrero and J. J. Velázquez, Chemotactic collapse for the Keller-Segel model, J. Math. Biol. 35, 177 (1996).
- G. Arumugam and J. Tyagi, Keller-Segel chemotaxis models: A review, Acta Appl. Math. 171, 6 (2021).
- J. E. Guyer, D. Wheeler, and J. A. Warren, FiPy: Partial differential equations with Python, Computi. Sci. Eng. 11, 6 (2009).
- https://github.com/mstaddon/phyllotaxis_keller_segel.
