The tensor-based model for growth and cell divisions of the root apex. I. The significance of principal directions


Plant organs grow symplastically, i.e. in a continuous and coordinated way. Such growth is of a tensor nature, which is manifested in the property that at every point of the organ three mutually orthogonal principal growth directions (PDG) can be recognized. The PDGs are postulated to affect orientation of cell divisions. This paper shows for the first time the 2D simulation model for growth in which cells divide taking into account the PDGs. The model, conceptually based on the growth tensor (GT), is applied to the root apex of radish, having a quiescent centre (QC). It shows the simulation of how exemplary cell pattern of the real root apex develops in time. The results provide satisfactory description of the root growth. The computer-generated cell pattern is realistic and more or less steady indicating that PDGs are important for growth. Presumably cells detect PDGs and obey them in the course of cell divisions. Computer generated division walls, perpendicular to PDGs, form periclinal and anticlinal zigzags as regular as those observed in microscopic sections. Growth tensor defines a field of growth rates at the organ level. QC, fundamental in this field, determines the group of quiescent initial cells which is, in turn, surrounded by active functional initials, from which all tissues of the root apex originate. The present simulations have shown that stability of generated cell pattern depends on whether the group of the functional initials is permanent; if this is not the case, the cell wall pattern changes in accordance with PDGs.

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Growth tensor


Principal growth direction


Principal stress direction


Quiescent centre


Relative elemental rate of linear growth


Two dimensional


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I thank Krzysztof Kurczynski and Jerzy Karczewski for their help in preparing the modelling software, and Zygmunt Hejnowicz and Dorota Kwiatkowska for discussion and critical reading of the manuscript.

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Correspondence to Jerzy Nakielski.

Electronic supplementary material

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Video S1a: Cell packet formation simulated taking into account the PDGs (AVI 927 kb)

Video S1a: Cell packet formation simulated taking into account the PDGs (AVI 927 kb)

Video S1b: Cell packet formation in which PDGs are not taken into account (AVI 717 kb)

Video S1b: Cell packet formation in which PDGs are not taken into account (AVI 717 kb)

Video S2: Simulation of the root apex growth - cells divide with respect to PDGs (AVI 944 kb)

Video S2: Simulation of the root apex growth - cells divide with respect to PDGs (AVI 944 kb)

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Nakielski, J. The tensor-based model for growth and cell divisions of the root apex. I. The significance of principal directions. Planta 228, 179–189 (2008) doi:10.1007/s00425-008-0728-y

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  • Growth simulation
  • Cell divisions
  • Principal directions
  • Root apex growth
  • Tensor approach