神经系统轴突三维定向生长的数值模拟

目的研究轴突在三维组织中的定向生长问题,探讨轴突沿非平坦基底定向生长时基底的低起伏程度对轴突的生长速率、成束和解束速率的影响。方法根据实验观察,设轴突生长的牵引力与靶细胞分泌的可扩散吸引分子的浓度梯度成正比,轴突成束和解束的侧向力与生长锥分泌的可扩散吸引分子和排斥分子的浓度梯度成正比,并且,吸引力指向浓度高的方向,排斥力指向浓度低的方向,浓度满足扩散方程。对于非平坦基底,有效力为沿基底轮廓线的切向分量。数值计算采用三维有限差分法和改进的Euler法。结果(1)轴突在三维组织中定向生长的基本特征与在二维培养基实验中的观察是一致的,只是在形态上有三维与二维之分;(2)非平坦基底的低起伏程度影响轴突的生长,随着起伏加剧,生长锥的前进速率减小、侧向速率增加。结论(1)许多基于二维培养实验所揭示的轴突生长的基本规律在三维情况下仍然是成立的;(2)基底的几何性质是影响轴突生长速率的重要因素之一。

3D numerical simulation of orientated growth of axons in the nervous system

Objective To study the orientated growth of axons in three-dimensional (3D) tissues, and to explore the effects on the rates of advancing, bundling and de-bundling of the axonal growth on an uneven surface of the extracellular matrices (ECM). Methods Based on the experimental observations, it is assumed that the traction for the axonal growth is proportional to the gradient of concentration of the diffusible attractants secreted by the target cells, and the lateral bundling or de-bundling force of the axons is proportional to the gradient of concentration which is the diffusible attractants or repellents secreted by the growth cones. The attractive force points to the uphill direction of the concentration field, the repulsive force points to the downhill direction, and the concentration itself satisfies the diffusion equation. When the growth cones crawl on an uneven surface of the ECM, only the tangential components of the above mentioned forces are helpful for the migration. In this numerical computing, 3D finite difference method and the improved Euler method were employed. Results (1) The basic characteristics of the orientated growth of axons in 3D tissues are similar to those in the 2D medium that were observed by the experiments, it is merely that they are morphologically termed by 2D or 3D problems. (2) The uneven of the ECM does affect the axonal growth. With the slope increasing, the advancing speed of the growth decreases and the lateral speed increases. Conclusions (1) Many mechanisms of the axonal growth revealed by the 2D culture experiments may be still valid in the corresponding 3D cases; (2) The geometry of the ECM is one of the key factors that have impacts on the rate of the axonal growth.