Growth tensor of plant organs

Abstract

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.