Theoretical Biology and Bioinformatics, Utrecht University, Utrecht, The Netherlands.
PLoS One. 2019 Aug 12;14(8):e0221059. doi: 10.1371/journal.pone.0221059. eCollection 2019.
We present a discrete mechanical model to study plant development. The method is built up of mass points, springs and hinges mimicking the plant cell wall's microstructure. To model plastic growth the resting lengths of springs are adjusted; when springs exceed a threshold length, new mass points, springs and hinges, are added. We formulate a stiffness tensor for the springs and hinges as a function of the fourth rank tensor of elasticity and the geometry of the mesh. This allows us to approximate the material law as a generalized orthotropic Hooke's law, and control material properties during growth. The material properties of the model are illustrated in numerical simulations for finite strain and plastic growth. To solve the equations of motion of mass points we assume elastostatics and use Verlet integration. The method is demonstrated in simulations when anisotropic growth causes emergent residual strain fields in cell walls and a bending of tissue. The method can be used in multilevel models to study plant development, for example by coupling it to models for cytoskeletal, hormonal and gene regulatory processes.
我们提出了一个离散力学模型来研究植物发育。该方法由质点、弹簧和铰链组成，模拟植物细胞壁的微观结构。为了模拟塑性生长，我们调整了弹簧的静长度；当弹簧超过阈值长度时，会添加新的质点、弹簧和铰链。我们将弹簧和铰链的弹性张量公式化为弹性的四阶张量和网格几何的函数。这使我们能够将材料定律近似为广义各向异性胡克定律，并在生长过程中控制材料性能。该模型的材料性能在有限应变和塑性生长的数值模拟中得到了说明。为了解决质点的运动方程，我们假设弹性静力学并使用 Verlet 积分。该方法在模拟中得到了验证，当各向异性生长导致细胞壁中出现残余应变场和组织弯曲时。该方法可用于多级模型来研究植物发育，例如通过将其与细胞骨架、激素和基因调控过程的模型耦合。
