[MS15] Toward creation of mathematical models by which treasures can be found in developmental biology

[MS15-02] Material orientation optimization model to explain vertebral growth via adaptation to mechanical loads

*Misaki Sakashita¹, Shin Sato¹, Yuta Nagai¹, Naoyuki Wada¹ (1. Tokyo University of Science (Japan))

Keywords:

bone formation,teleost fish,topological defect

This study aims to explain how the material orientation of bone adapts to mechanical loads and how this orientation influences bone formation. Bone is an anisotropic material with an orientation formed by the deposition of hydroxyapatite, a highly anisotropic crystal, along collagen fibers. The well-aligned collagen fibers and hydroxyapatite allow bone to withstand mechanical loads along the orientations. Our previous observations have shown that the fiber orientation on the vertebral surface of fish changes from late larval to adult stages. During this period, the random orientation on the lateral sides of the vertebrae changes to an anteroposterior orientation, whereas the orientations on the anterior and posterior ends remain dorsoventral. Also, bony projections are formed at the anterior and posterior ends of the lateral sides and extend toward the center to form plate-like ridges that support the vertebrae in the anteroposterior direction. Because topological defects in fiber orientation, which are singularity points where the orientation order is lost, are located at the starting point of morphogenesis, such as tissue elongation and aggregation, we hypothesized that the fiber orientation of vertebrae adapts to mechanical loads and that topological defects function as the starting points for the formation of lateral ridges. We built a simple two-dimensional orientation optimization model to reproduce the fiber orientation on the lateral side of fish vertebrae. In this optimization model, the objective function was a weighted sum of Landau-de Gennes free energy and strain energy density. By minimizing this objective function, the fiber orientation is aligned to obtain high material stiffness. We set the dorsoventral orientation at the anterior and posterior ends of the vertebrae as the boundary condition and applied the anteroposterior compressive loads to these ends. The numerical result showed that the orientation on the lateral sides was aligned anteroposteriorly, whereas the orientation at the ends was dorsoventral. In addition, a topological defect appeared at two points near the vertebral ends where the anteroposterior and dorsoventral orientations switched. To evaluate the numerical results, we further analyzed the orientation of the mineralized bone of zebrafish vertebrae in detail. The mineralized bone was oriented dorsoventrally at the vertebral ends and anteroposteriorly on the lateral sides. Also, abrupt orientation changes of the mineralized bone, similar to topological defects, were found at the base of the plate-like ridges. Because the orientation analysis matches the numerical results, our mathematical model can simulate the mechanical adaptation of bone material orientation and predict the starting point of morphogenesis in fish vertebrae.