Modeling Plant Tissue Growth and Cell Division



Morphogenesis is the creation of form, a complex process requiring the integration of genetics, mechanics, and geometry. Patterning processes driven by molecular regulatory and signaling networks interact with growth to create organ shape, often in unintuitive ways. Computer simulation modeling is becoming an increasingly important tool to aid our understanding of these complex interactions. In this chapter we introduce computational approaches for studying these processes on spatial, multicellular domains. For some problems, such as the exploration of many patterning processes, simulation can be done on static (non-growing) templates. These can range from abstract idealized cells, such as rectangular or hex grids, to more realistic shapes such as Voronoi regions, or even shapes extracted from bio-imaging data. More dynamic processes like phyllotaxis involve the interaction of growth and patterning, and require the simulation of growing domains. In the simplest case growth can be modeled descriptively, provided as an input to the model. Growth is specified globally, and must be designed carefully to avoid conflicts (growing cells must fit together). We present several methods for this that can be applied to shoots, roots, leaves, and other plant organs. However when shape is an emergent property of the model, different cells or areas of the tissue need to specify their growth locally, and physically-based methods (mechanics) are required to resolve conflicts. Among these are mass-spring, finite element, and Hamiltonian-based approaches.


Growth Tensor Parametric Coordinates Rest Length Cell Wall Segments Symplastic Growth 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Funding is gratefully acknowledged from the Bundesministerium für Bildung und Forschung grants 031A492 and 031A494, the Swiss National Science Foundation Plant Growth RTD, Human Frontiers Science Program grant RGP0008/2013 to R.S.S., Marie Skodowska-Curie individual fellowship (Horizon 2020, 703886) to A.R., and the Max Planck Institute for Plant Breeding Research, Cologne, Germany. Some sections of this chapter were adapted from the lecture notes of the Les Houches summer school of 2009 [75]. We would also like to acknowledge Przemyslaw Prusinkiewicz and the members of his lab for helping to formulate many of the ideas appearing in this chapter.


  1. 1.
    Abley K, De Reuille PB, Strutt D, Bangham A, Prusinkiewicz P, Marée AF, Grieneisen VA, Coen E (2013) An intracellular partitioning-based framework for tissue cell polarity in plants and animals. Development 140(10):2061–2074CrossRefGoogle Scholar
  2. 2.
    Armour WJ, Barton DA, Law AM, Overall RL (2015) Differential growth in periclinal and anticlinal walls during lobe formation in arabidopsis cotyledon pavement cells. Plant Cell 27(9):2484–2500CrossRefGoogle Scholar
  3. 3.
    Bassel GW, Stamm P, Mosca G, de Reuille PB, Gibbs DJ, Winter R, Janka A, Holdsworth MJ, Smith RS (2014) Mechanical constraints imposed by 3d cellular geometry and arrangement modulate growth patterns in the arabidopsis embryo. Proc Natl Acad Sci 111(23):8685–8690CrossRefGoogle Scholar
  4. 4.
    Besson S, Dumais J (2011) Universal rule for the symmetric division of plant cells. Proc Natl Acad Sci 108(15):6294–6299CrossRefGoogle Scholar
  5. 5.
    Bilsborough GD, Runions A, Barkoulas M, Jenkins HW, Hasson A, Galinha C, Laufs P, Hay A, Prusinkiewicz P, Tsiantis M (2011) Model for the regulation of arabidopsis thaliana leaf margin development. Proc Natl Acad Sci 108(8):3424–3429CrossRefGoogle Scholar
  6. 6.
    Boudon F, Pradal C, Cokelaer T, Prusinkiewicz P, Godin C (2012) L-py: an l-system simulation framework for modeling plant architecture development based on a dynamic language. Front. Plant Sci. 3:76CrossRefGoogle Scholar
  7. 7.
    Boudon F, Chopard J, Ali O, Gilles B, Hamant O, Boudaoud A, Traas J, Godin C (2015) A computational framework for 3d mechanical modeling of plant morphogenesis with cellular resolution. PLoS Comput Biol 11(1):e1003950CrossRefGoogle Scholar
  8. 8.
    Campilho A, Garcia B, Wijk HV, Campilho A, Scheres B et al (2006) Time-lapse analysis of stem-cell divisions in the arabidopsis thaliana root meristem. Plant J 48(4):619–627CrossRefGoogle Scholar
  9. 9.
    Cieslak M, Runions A, Prusinkiewicz P (2015) Auxin-driven patterning with unidirectional fluxes. J Exp Bot 66:5083–5102. CrossRefGoogle Scholar
  10. 10.
    Coen E, Rebocho AB (2016) Resolving conflicts: modeling genetic control of plant morphogenesis. Dev Cell 38(6):579–583CrossRefGoogle Scholar
  11. 11.
    Coen E, Rolland-Lagan AG, Matthews M, Bangham JA, Prusinkiewicz P (2004) The genetics of geometry. Proc Natl Acad Sci USA 101(14):4728–4735CrossRefGoogle Scholar
  12. 12.
    de Boer MJ, Fracchia FD, Prusinkiewicz P (1992) A model for cellular development in morphogenetic fields. In: Lindenmayer systems. Springer, Berlin, pp 351–370CrossRefGoogle Scholar
  13. 13.
    de Reuille PB, Routier-Kierzkowska AL, Kierzkowski D, Bassel GW, Schüpbach T, Tauriello G, Bajpai N, Strauss S, Weber A, Kiss A et al (2015) Morphographx: a platform for quantifying morphogenesis in 4d. Elife 4:e05864CrossRefGoogle Scholar
  14. 14.
    De Rybel B, Adibi M, Breda AS, Wendrich JR, Smit ME, Novák O, Yamaguchi N, Yoshida S, Van Isterdael G, Palovaara J et al (2014) Integration of growth and patterning during vascular tissue formation in arabidopsis. Science 345(6197):1255215CrossRefGoogle Scholar
  15. 15.
    Donnelly P, Bonetta D, Tsukaya H, Dengler R, Dengler N (1999) Cell cycling and cell enlargement in developing leaves of Arabidopsis. Dev Biol 215(2):407–419CrossRefGoogle Scholar
  16. 16.
    Dupuy L, Mackenzie J, Haseloff J (2010) Coordination of plant cell division and expansion in a simple morphogenetic system. Proc Natl Acad Sci 107(6):2711–2716CrossRefGoogle Scholar
  17. 17.
    el Showk S, Blomster T, Siligato R, Marée AF, Mähönen AP, Grieneisen VA et al (2015) Parsimonious model of vascular patterning links transverse hormone fluxes to lateral root initiation: auxin leads the way, while cytokinin levels out. PLoS Comput Biol 11(10):e1004450CrossRefGoogle Scholar
  18. 18.
    Errera L (1888) Über zellfromen und seifenblasen. Botanisches Centralblatt 34:395–398Google Scholar
  19. 19.
    Federl P, Prusinkiewicz P (1999) Virtual laboratory: an interactive software environment for computer graphics. In: Computer graphics international, vol 242, pp 93–100Google Scholar
  20. 20.
    Fernandez R, Das P, Mirabet V, Moscardi E, Traas J, Verdeil JL, Malandain G, Godin C (2010) Imaging plant growth in 4d: robust tissue reconstruction and lineaging at cell resolution. Nat Methods 7(7):547–553CrossRefGoogle Scholar
  21. 21.
    Feugier FG, Mochizuki A, Iwasa Y. (2005) Self-organization of the vascular system in plant leaves: inter-dependent dynamics of auxin flux and carrier proteins. J Theor Biol 236(4):366–375CrossRefGoogle Scholar
  22. 22.
    Fukushima K, Fujita H, Yamaguchi T, Kawaguchi M, Tsukaya H, Hasebe M (2015) Oriented cell division shapes carnivorous pitcher leaves of sarracenia purpurea. Nat Commun 6:6450CrossRefGoogle Scholar
  23. 23.
    Galassi M, Davies J, Theiler J, Gough B, Jungman G, Alken P, Booth M, Rossi F (2002) Gnu Scientific Library. Network Theory Ltd 3Google Scholar
  24. 24.
    Giavitto JL, Michel O (2001) MGS: a ruled-based language for complex objects and collections. Electron Notes Theor Comput Sci 59(4):1–19CrossRefGoogle Scholar
  25. 25.
    Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetik 12(1):30–39CrossRefGoogle Scholar
  26. 26.
    Glazier JA, Graner F (1993) Simulation of the differential adhesion driven rearrangement of biological cells. Phys Rev E 47(3):2128CrossRefGoogle Scholar
  27. 27.
    Goriely A, Robertson-Tessi M, Tabor M, Vandiver R (2008) Elastic growth models. In: Mathematical modelling of biosystems. Springer, Berlin, pp 1–44Google Scholar
  28. 28.
    Grieneisen VA, Xu J, Marée AFM, Hogeweg P, Scheres B (2007) Auxin transport is sufficient to generate a maximum and gradient guiding root growth. Nature 449(7165):1008–1013. CrossRefGoogle Scholar
  29. 29.
    Haselkorn R (1998) How cyanobacteria count to 10. Science 282(5390):891–892CrossRefGoogle Scholar
  30. 30.
    Heisler MG, Jönsson H (2006) Modeling auxin transport and plant development. J Plant Growth Regul 25:302–312. CrossRefGoogle Scholar
  31. 31.
    Hejnowicz Z, Karczewski J (1993) Modeling of meristematic growth of root apices in a natural coordinate system. Am J Bot 80:309–315CrossRefGoogle Scholar
  32. 32.
    Hejnowicz Z, Nakielski J, Hejnowicz K (1984) Modeling of spatial variations of growth within apical domes by means of the growth tensor. ii. Growth specified on dome surface. Acta Soc Bot Pol 53:301–316.CrossRefGoogle Scholar
  33. 33.
    Hervieux N, Dumond M, Sapala A, Routier-Kierzkowska AL, Kierzkowski D, Roeder AH, Smith RS, Boudaoud A, Hamant O (2016) A mechanical feedback restricts sepal growth and shape in arabidopsis. Curr Biol 26(8):1019–1028CrossRefGoogle Scholar
  34. 34.
    Hofmeister W (1868) Handbuch der physiologishen botanik. Engelmann, LeipzigGoogle Scholar
  35. 35.
    Honda H (1978) Description of cellular patterns by Dirichlet domains: the two-dimensional case. J Theor Biol 72:523–543CrossRefGoogle Scholar
  36. 36.
    Honda H (1983) Geometrical models for cells in tissues. Int Rev Cytol 81:191–248CrossRefGoogle Scholar
  37. 37.
    Jönsson H, Heisler MG, Shapiro BE, Meyerowitz EM, Mjolsness E (2006) An auxin-driven polarized transport model for phyllotaxis. Proc Natl Acad Sci U S A 103(5):1633–1638. CrossRefGoogle Scholar
  38. 38.
    Jönsson H, Gruel J, Krupinski P, Troein C (2012) On evaluating models in computational morphodynamics. Curr Opin Plant Biol 15(1):103–110CrossRefGoogle Scholar
  39. 39.
    Karlebach G, Shamir R (2008) Modelling and analysis of gene regulatory networks. Nat Rev Mol Cell Biol 9(10):770–780CrossRefGoogle Scholar
  40. 40.
    Kennaway R, Coen E, Green A, Bangham A (2011) Generation of diverse biological forms through combinatorial interactions between tissue polarity and growth. PLoS Comput Biol 7(6):e1002071CrossRefGoogle Scholar
  41. 41.
    Kierzkowski D, Nakayama N, Routier-Kierzkowska AL, Weber A, Bayer E, Schorderet M, Reinhardt D, Kuhlemeier C, Smith RS (2012) Elastic domains regulate growth and organogenesis in the plant shoot apical meristem. Science 335(6072):1096–1099CrossRefGoogle Scholar
  42. 42.
    Kramer EM (2008) Computer models of auxin transport: a review and commentary. J Exp Bot 59(1):45–53CrossRefGoogle Scholar
  43. 43.
    Kramer EM (2009) Auxin-regulated cell polarity: an inside job? Trends Plant Sci 14(5):242–247CrossRefGoogle Scholar
  44. 44.
    Kuchen EE, Fox S, de Reuille PB, Kennaway R, Bensmihen S, Avondo J, Calder GM, Southam P, Robinson S, Bangham A et al (2012) Generation of leaf shape through early patterns of growth and tissue polarity. Science 335(6072):1092–1096CrossRefGoogle Scholar
  45. 45.
    Kuhlemeier C (2007) Phyllotaxis. Trends Plant Sci 12(4):143–150CrossRefGoogle Scholar
  46. 46.
    Kwiatkowska D (2006) Flower primordium formation at the arabidopsis shoot apex: quantitative analysis of surface geometry and growth. J Exp Bot 57(3):571–580CrossRefGoogle Scholar
  47. 47.
    Lindenmayer A (1968) Mathematical models for cellular interactions in development. I. Filaments with one-sided inputs. J Theor Biol 18(3):280–299CrossRefGoogle Scholar
  48. 48.
    Lindenmayer A (1968) Mathematical models for cellular interactions in development. II. Simple and branching filaments with two-sided inputs. J Theor Biol 18(3):300–315PubMedGoogle Scholar
  49. 49.
    Lintilhac PM, Vesecky TB (1984) Stress-induced alignment of division plane in plant tissues grown in vitro. Nature 307(5949):363–364CrossRefGoogle Scholar
  50. 50.
    Lockhart JA (1965) An analysis of irreversible plant cell elongation. J Theor Biol 8(2):264–275CrossRefGoogle Scholar
  51. 51.
    Louveaux M, Julien JD, Mirabet V, Boudaoud A, Hamant O (2016) Cell division plane orientation based on tensile stress in arabidopsis thaliana. Proc Natl Acad Sci 113(30):E4294–303. CrossRefGoogle Scholar
  52. 52.
    Meinhardt H (1982) Models of biological pattern formation. Academic Press, LondonGoogle Scholar
  53. 53.
    Meinhardt H (2003) Complex pattern formation by a self-destabilization of established patterns: chemotactic orientation and phyllotaxis as examples. C R Biol 326(2):223–237CrossRefGoogle Scholar
  54. 54.
    Merks RM, Guravage M, Inzé D, Beemster GT (2011) Virtualleaf: an open-source framework for cell-based modeling of plant tissue growth and development. Plant Physiol 155(2):656–666CrossRefGoogle Scholar
  55. 55.
    Mitchison GJ (1980) A model for vein formation in higher plants. Philos Trans R Soc Lond B Biol Sci 207:79–109CrossRefGoogle Scholar
  56. 56.
    Nakielski J (2000) Pattern formation in biology, vision and dynamics, chap. Tensorial model for growth and cell division in the shoot apex. World Scientific, pp. 252–286Google Scholar
  57. 57.
    Nakielski J, Barlow P (1995) Principal directions of growth and the generation of cell patterns in wild-type and gib-1 mutant roots of tomato (lycopersicon esculentum mill.) grown in vitro. Planta 196(1):30–39CrossRefGoogle Scholar
  58. 58.
    Nakielski J, Lipowczan M (2013) Spatial and directional variation of growth rates in arabidopsis root apex: A modelling study. PLOS ONE 8(12).
  59. 59.
    Neubert MG, Caswell H, Murray J (2002) Transient dynamics and pattern formation: reactivity is necessary for turing instabilities. Math Biosci 175(1):1–11CrossRefGoogle Scholar
  60. 60.
    Prusinkiewicz P, Lane B (2012) Pattern formation in morphogenesis. Springer, BerlinGoogle Scholar
  61. 61.
    Prusinkiewicz P, Lane B (2013) Modeling morphogenesis in multicellular structures with cell complexes and l-systems. In: Pattern formation in morphogenesis. Springer, Berlin, pp 137–151CrossRefGoogle Scholar
  62. 62.
    Prusinkiewicz P, Lindenmayer A (1990) Algorithmic beauty of plants. Springer, BerlinCrossRefGoogle Scholar
  63. 63.
    Rebocho AB, Southam P, Kennaway JR, Bangham JA, Coen E (2017) Generation of shape complexity through tissue conflict resolution. eLife 6:e20156CrossRefGoogle Scholar
  64. 64.
    Reinhardt D, Pesce ER, Stieger P, Mandel T, Baltensperger K, Bennett M, Traas J, Friml J, Kuhlemeier C (2003) Regulation of phyllotaxis by polar auxin transport. Nature 426(6964):255–260. CrossRefGoogle Scholar
  65. 65.
    Rodriguez EK, Hoger A, McCulloch AD (1994) Stress-dependent finite growth in soft elastic tissues. J Biomech 27(4):455–467CrossRefGoogle Scholar
  66. 66.
    Rolland-Lagan AG, Prusinkiewicz P (2005) Reviewing models of auxin canalization in the context of leaf vein pattern formation in Arabidopsis. Plant J 44(5):854–865. CrossRefGoogle Scholar
  67. 67.
    Rolland-Lagan AG, Remmler L, Girard-Bock C (2014) Quantifying shape changes and tissue deformation in leaf development. Plant Physiol 165(2):496–505CrossRefGoogle Scholar
  68. 68.
    Runions A (2008) Modeling biological patterns using the space colonization algorithm. M.Sc. Thesis, University of CalgaryGoogle Scholar
  69. 69.
    Runions A, Fuhrer M, Lane B, Federl P, Rolland-Lagan AG, Prusinkiewicz P (2005) Modeling and visualization of leaf venation patterns. ACM Trans Graph 24:702–711CrossRefGoogle Scholar
  70. 70.
    Sachs T (1981) The control of patterned differentiation of vascular tissues. Adv Bot Res 9:151–262CrossRefGoogle Scholar
  71. 71.
    Sahlin P, Söderberg B, Jönsson H (2009) Regulated transport as a mechanism for pattern generation: capabilities for phyllotaxis and beyond. J Theor Biol 258(1):60–70CrossRefGoogle Scholar
  72. 72.
    Sauret-Güeto S, Schiessl K, Bangham A, Sablowski R, Coen E (2013) Jagged controls arabidopsis petal growth and shape by interacting with a divergent polarity field. PLoS Biol 11(4):e1001550CrossRefGoogle Scholar
  73. 73.
    Scarpella E, Francis P, Berleth T (2004) Stage-specific markers define early steps of procambium development in Arabidopsis leaves and correlate termination of vein formation with mesophyll differentiation. Development 131(14):3445–3455CrossRefGoogle Scholar
  74. 74.
    Scarpella E, Marcos D, Friml J, Berleth T (2006) Control of leaf vascular patterning by polar auxin transport. Genes Dev 20(8):1015–1027. CrossRefGoogle Scholar
  75. 75.
    Smith R (2011) Modeling plant morphogenesis and growth. New Trends Phys Mech Biol Syst 92:301–336CrossRefGoogle Scholar
  76. 76.
    Smith RS, Bayer EM (2009) Auxin transport-feedback models of patterning in plants. Plant Cell Environ 32(9): 1258–1271. CrossRefGoogle Scholar
  77. 77.
    Smith RS, Guyomarc’h S, Mandel T, Reinhardt D, Kuhlemeier C, Prusinkiewicz P (2006) A plausible model of phyllotaxis. Proc Natl Acad Sci U S A 103(5):1301–1306. CrossRefGoogle Scholar
  78. 78.
    Smith RS, Kuhlemeier C, Prusinkiewicz P (2006) Inhibition fields for phyllot actic pattern formation: a simulation study. Can J Bot 84(11):1635–1649CrossRefGoogle Scholar
  79. 79.
    Turing A (1952) The chemical basis of morphogenesis. Philos Trans R Soc Lond B Biol Sci 237:37–52CrossRefGoogle Scholar
  80. 80.
    Wabnik K, Robert HS, Smith RS, Friml J (2013) Modeling framework for the establishment of the apical-basal embryonic axis in plants. Curr Biol 23(24):2513–2518CrossRefGoogle Scholar
  81. 81.
    Yoshida S, de Reuille PB, Lane B, Bassel GW, Prusinkiewicz P, Smith RS, Weijers D (2014) Genetic control of plant development by overriding a geometric division rule. Dev cell 29(1):75–87CrossRefGoogle Scholar
  82. 82.
    Žádníková P, Wabnik K, Abuzeineh A, Gallemi M, Van Der Straeten D, Smith RS, Inzé D, Friml J, Prusinkiewicz P, Benková E (2016) A model of differential growth-guided apical hook formation in plants. Plant Cell 28(10):2464–2477CrossRefGoogle Scholar
  83. 83.
    Zienkiewicz OC, Taylor RL (2005) The finite element method for solid and structural mechanics. Butterworth-Heinemann, BostonGoogle Scholar

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Authors and Affiliations

  1. 1.Max Planck Institute for Plant Breeding ResearchKölnGermany

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