Zygmunt Hejnowicz

Plant Structure: Function and Development

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Growth tensor of plant organs

Analysis of the definition of the relative elemental rate of growth of a line element, RERG1, in a growing plant organ leads to the dyadic ∇V where V is the vector field of displacement velocities of material points in the organ. The components of this dyadic represent physical components of a tensor, which we propose to call the growth tensor. The latter can be derived directly from the definition of RERG1. The growth tensor allows full characterization of the rate of growth in length, area, and volume, as well as rates of angular change between elements, and of vorticity in the growing organ. From the fact that anticlinal and periclinal walls of cells within the organ preserve their orthogonality during growth, we infer that the principal directions of the growth tensor coincide with periclines and anticlines. The definition of the growth tensor based on the dyadic offers an easy way to generate this tensor in different coordinate systems. An example is given of the use of the growth tensor in analyzing elongation growth of a cylindrical plant organ in two alternative modes: with and without rotation of the tip. It is shown that growth by the two modes yields the same relative elemental rates of growth in volume, but that the principal directions of the growth tensors are different. We infer that if growth is a tensorial attribute of an organ, then the controls of growth must also be tensorial attributes. The controlling tensors must have at least as high a rank as the growth tensors, but must be of a higher hierarchical level.

Calculation of rates of cell division in shoot meristems with reference to the growth tensor

The theoretical concept of a cell division tensor has some appeal; yet, the actual formulation of such a tensor remains a challenge. It is nonetheless possible, without invoking a cell division tensor, to use the physical components of the growth tensor, along with natural curvilinear coordinate systems, to calculate the theoretical distribution of cell division rates within apical meristems of plant shoots. The patterns of dimensional changes of cells, the distribution of growth rates specified by the growth tensor, and the indicated rates of cell division are described for meristems characterized by steady state fields of cell flux velocities and of mean cell dimensions. These relations are evaluated by means of continuity equations for the fluxes of mean cell dimensions. These methods allow calculation of the relative rates of cell division at specified sites, and within specified orientation classes, as well as of division rates irrespective of orientation.