Anisotropic plant cell elongation due to ortho-gravitropism

In this article we elucidate the well-known biological phenomenon (geotropism) as governed by physical mechanisms, resulting from internal biochemical reactions, in terms of mathematics. Gravitropism causes vertical orientation of plant's axis and in its special cases of positive (root) and negative (stem) geotropism together is called ortho-geotropism. It represents one of the most rapid and visually obvious response of plants to the influence of gravitational field. Seeking for approximate description for this phenomenon we confine to a single cell approach and we begin with the Lockhart equation considering a plant cell as a homogeneous one. In principle, the latter should also account for the existing anisotropies due to mechanical stresses (auxin redistribution). Hence, all global quantities like internal pressure or turgor threshold become direction dependent and consequently acquire tensor representation. Moreover, by involving explicitly time dependence the tensor differential equation becomes a dynamic one. In the context of ortho-geotropism, where gravitational field causes movement of phytohormones and mobile particles following gravity (statolith theory) a basic solution of our tensor equation is found and detailed step by step derivations are presented. By considering only positive (root) geotropism we may, however, extend our solution to the stem bending even though the biological mechanisms differ. Both solutions represent two possible empirical situations which have been probed and verified worldwide ever since.