[论文解读] Dynamics of Plant Growth; A Theory Based on Riemannian Geometry

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

Plant Molecular Biology Research参考文献 48被引用 3

一句话总结

本文提出了一种基于黎曼几何的植物组织生长模型，将一维和二维组织视为连续、可变形的曲面，受耦合张量方程控制，这些方程将生长与曲率动力学相耦合。该模型再现了关键的生物生长模式，包括根中从单峰到双峰的相对单元生长速率的转变，以及二维圆盘中负曲率和波纹状结构的出现，为宏观植物形态发生提供了一个统一的几何框架。

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.

研究动机与目标

• 开发一种超越经典欧几里得模型的宏观植物组织生长几何框架。
• 解释根中相对单元生长速率（REGR）的时间动力学，包括在高分辨率研究中观察到的从单峰到双峰结构的转变。
• 在圆形圆盘和各向异性表面等条件下，对二维组织生长及其曲率演化进行建模，作为植物叶片和针状藻帽的代理。
• 通过度量、速度场和形变张量（扩张、剪切、旋转）等动力学变量表征组织变形。
• 提供一个统一的、物理上合理的生长模型，整合曲面空间中的反应-扩散过程与物质源动力学。

提出的方法

• 将一维和二维植物组织建模为非欧几里得（弯曲）黎曼空间中的连续、可变形几何体。
• 建立一组耦合张量方程，通过曲率与物质添加之间的反馈机制描述生长动力学。
• 在几何框架中将生长同时视为反应-扩散过程和新物质的分布源。
• 使用数值模拟求解在各种对称性和各向异性条件下的一维根和二维圆盘的张量方程。
• 计算度量张量、速度场以及形变张量（扩张、剪切、旋转）等动力学变量，以表征组织演化。
• 将模型预测与薄圆盘系统及高分辨率根生长研究的实验数据进行比较。

实验结果

研究问题

• RQ1 如何在弯曲的非欧几里得空间中将植物组织生长建模为几何过程？
• RQ2 模型中何种机制解释了根中相对单元生长速率（REGR）从单峰到双峰的转变？
• RQ3 在二维生长组织中，曲率如何演化？这能否解释叶片状结构中观察到的负曲率？
• RQ4 该模型能否预测二维曲面上复杂形态（如波纹状和伸长）的出现？
• RQ5 形变张量（扩张、剪切、旋转）在表征组织生长动力学中扮演何种角色？

主要发现

• 该模型通过在弯曲几何框架中模拟模块化生长动力学，再现了一维根生长的伸长区。
• 在仿真中，从单峰到双峰的REGR转变可直接归因于弯曲空间中反应-扩散过程与分布物质源动力学的相互作用。
• 在二维圆形圆盘中，模型预测了负曲率的出现，与用作植物叶片代理的薄圆盘系统实验观察结果一致。
• 对于各向异性的二维圆盘，模型预测了具有局部波纹状结构的伸长形貌，表明存在一种几何机制可解释形态复杂性。
• 度量张量和速度场的时间演化，连同形变张量，完整表征了组织生长的动力学，包括扩张、剪切和旋转分量。
• 该模型建立了几何反馈（曲率与生长）与实验观测到的生长动力学之间的直接联系，为植物形态发生提供了新颖的理论基础。