Dynamics of Plant Growth; A Theory Based on Riemannian Geometry

Abstract

In this work, a new model for macroscopic plant tissue growth based on dynamical Riemannian geometry is presented. We treat 1D and 2D tissues as continuous, deformable, growing geometries for sizes larger than 1mm. The dynamics of the growing tissue are described by a set of coupled tensor equations in non-Euclidean (curved) space. These coupled equations represent a novel feedback mechanism between growth and curvature dynamics. For 1D growth, numerical simulations are compared to two measures of root growth. First, modular growth along the simulated root shows an elongation zone common to many species of plant roots. Second, the relative elemental growth rate (REGR) calculated in silico exhibits temporal dynamics recently characterized in high-resolution root growth studies but which thus far lack a biological hypothesis to explain them. Namely, the REGR can evolve from a single peak localized near the root tip to a double-peak structure. In our model, this is a direct consequence of considering growth as both a geometric reaction-diffusion process and expansion due to a distributed source of new materials. In 2D, we study a circularly symmetric growing disk with emergent negative curvatures. These results are compared against thin disk experiments, which are a proxy model for plant leaves. These results also apply to the curvature evolution and the inhomogeneous growth pattern of the Acetabularia cap. Lastly, we extend the model to anisotropic disks and predict the growth dynamics for a 2D curved surface which develops an elongated shape with localized ruffling. Our model also provides several measures of the dynamics of tissue growth. These include the time evolution of the metric and velocity field, which are dynamical variables in the model, as well as expansion, shear and rotation which are deformation tensors characterizing the growth of the tissue. The latter are physically measurable quantities that remain to be fully explored using modern tissue growth imagining techniques.