Dynamic Equations of Axial Root Growth from the Empirical Solution of a Mechanistic Model

Abstract

An energy conservation model of axial plant growth is solved, and the solution equations are applied to root growth. The axial growth model is a system of coupled, first-order partial differential equations for the local tissue displacement velocities, V(X, t), and longitudinal tissue strain, E(X, t). The growth equations are solved in the material half-space, X ≥ 0, t ≥ 0, by the method of characteristics. For the applications presented here, the boundary conditions and the coefficient values are found from literature values of steady-state displacement velocities and cell formation rates. The model solution generates a dynamic description of axial growth from steady-state data and yields a more general description of growth. The theoretical spatial displacement velocities, v(x, t), accurately describe steady-state velocity measurements. The spatial strain, e(x, t), is employed to derive an equation for the distribution of cell lengths along an axis, l(x, t). The equation for cell lengths precisely states the contribution of cell division and cell elongation processes to the pattern of cell lengths on an axis. A comparison of the theoretical cell length curves to the experimental measurements yields an empirical validation of the model by accurately describing root-growth data collected independently from the input data to the model.