[Paper Review] Dynamics of Plant Growth; A Theory Based on Riemannian Geometry

Julia Pulwicki|arXiv (Cornell University)|Dec 22, 2015

Plant Molecular Biology Research48 references3 citations

TL;DR

This paper proposes a Riemannian geometric model of plant tissue growth, treating 1D and 2D tissues as continuous, deformable, curved surfaces governed by coupled tensor equations that couple growth to curvature dynamics. The model reproduces key biological growth patterns, including the transition from single- to double-peak relative elemental growth rates in roots and the emergence of negative curvature and ruffling in 2D disks, offering a unified geometric framework for macroscopic plant morphogenesis.

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.

Motivation & Objective

• To develop a geometric framework for macroscopic plant tissue growth beyond classical Euclidean models.
• To explain the temporal dynamics of relative elemental growth rate (REGR) in roots, including the transition from single- to double-peak structures observed in high-resolution studies.
• To model 2D tissue growth with curvature evolution, particularly in circular disks and anisotropic surfaces, as a proxy for plant leaves and Acetabularia caps.
• To characterize tissue deformation through dynamical variables like metric, velocity field, and deformation tensors (expansion, shear, rotation).
• To provide a unified, physically grounded model of growth that integrates reaction-diffusion and material source dynamics in curved space.

Proposed method

• Modeling 1D and 2D plant tissues as continuous, deformable geometries in non-Euclidean (curved) Riemannian space.
• Formulating a system of coupled tensor equations that describe growth dynamics through feedback between curvature and material addition.
• Incorporating growth as both a reaction-diffusion process and a distributed source of new material in the geometric framework.
• Using numerical simulations to solve the tensor equations for 1D roots and 2D disks under various symmetry and anisotropy conditions.
• Computing dynamical variables such as the metric tensor, velocity field, and deformation tensors (expansion, shear, rotation) to characterize tissue evolution.
• Comparing model predictions with experimental data from thin disk systems and high-resolution root growth studies.

Experimental results

Research questions

• RQ1 How can plant tissue growth be modeled as a geometric process in curved, non-Euclidean space?
• RQ2 What mechanism in the model explains the transition from single-peak to double-peak relative elemental growth rate (REGR) in roots?
• RQ3 How does curvature evolve in 2D growing tissues, and can this explain observed negative curvatures in leaf-like structures?
• RQ4 Can the model predict the emergence of complex morphologies such as ruffling and elongation in 2D curved surfaces?
• RQ5 What role do deformation tensors (expansion, shear, rotation) play in characterizing the dynamics of tissue growth?

Key findings

• The model reproduces the elongation zone in 1D root growth by simulating modular growth dynamics in a curved geometric framework.
• The transition from single- to double-peak REGR in silico is directly attributed to the interplay between reaction-diffusion and distributed material source dynamics in curved space.
• In 2D circular disks, the model predicts the emergence of negative curvature, consistent with experimental observations of thin disk systems used as proxies for plant leaves.
• For anisotropic 2D disks, the model predicts the development of elongated shapes with localized ruffling, suggesting a geometric mechanism for morphological complexity.
• The time evolution of the metric tensor and velocity field, along with deformation tensors, provides a complete dynamical characterization of tissue growth, including expansion, shear, and rotation components.
• The model establishes a direct link between geometric feedback (curvature and growth) and experimentally observed growth dynamics, offering a novel theoretical basis for plant morphogenesis.