Institute of Robotics and Automatic Information Systems, Nankai University, College of Artificial Intelligence, 201-02, Tianjin, 300350, People's Republic of China.
Tianjin Key Laboratory of Intelligent Robotics, Nankai University, Tianjin, 300350, People's Republic of China.
Sci Rep. 2021 Mar 1;11(1):4813. doi: 10.1038/s41598-021-84313-7.
The meshwork pattern is a significant pattern in the development of biological tissues and organs. It is necessary to explore the mathematical mechanism of meshwork pattern formation. In this paper, we found that the meshwork pattern is formed by four kinds of stalk behaviours: stalk extension, tip bifurcation, side branching and tip fusion. The Turing-type pattern underlying the meshwork pattern is a Turing spot pattern, which indicates that the Turing instability of the spot pattern promotes activator peak formation and then guides the formation of meshwork patterns. Then, we found that the Turing wavelength decreased in turn from tip bifurcation to side branching to tip fusion via statistical evaluation. Through the functional relationship between the Turing wavelength and model parameters ([Formula: see text] and [Formula: see text]), we found that parameters [Formula: see text] and [Formula: see text] had monotonic effects on the Turing wavelength and that parameter [Formula: see text] had nonmonotonic effects. Furthermore, we performed simulations of local meshwork pattern formation under variable model parameter values. The simulation results verified the corresponding relationship between the Turing wavelength and stalk behaviours and the functional relationship between the Turing wavelength and model parameters. The simulation results showed that the Turing wavelength regulated the meshwork pattern and that the small Turing wavelength facilitated dense meshwork pattern formation. Our work provides novel insight into and understanding of the formation of meshwork patterns. We believe that studies associated with network morphogenesis can benefit from our work.
网格图案是生物组织和器官发育的重要图案。有必要探索网格图案形成的数学机制。在本文中，我们发现网格图案是由四种茎干行为形成的：茎干延伸、尖端分叉、侧支和尖端融合。网格图案下的图灵型图案是图灵斑图案，这表明图灵不稳定性促进了激活剂峰的形成，然后引导了网格图案的形成。然后，我们通过统计评估发现，从尖端分叉到侧支再到尖端融合，图灵波长依次减小。通过图灵波长与模型参数之间的函数关系([Formula: see text] 和 [Formula: see text])，我们发现参数 [Formula: see text] 和 [Formula: see text] 对图灵波长有单调效应，而参数 [Formula: see text] 有非单调效应。此外，我们在可变模型参数值下进行了局部网格图案形成的模拟。模拟结果验证了图灵波长与茎干行为之间的对应关系以及图灵波长与模型参数之间的函数关系。模拟结果表明，图灵波长调节了网格图案，较小的图灵波长有利于密集的网格图案形成。我们的工作为网格图案的形成提供了新的见解和理解。我们相信与网络形态发生相关的研究可以从我们的工作中受益。
