Shoji Hiroto, Iwasa Yoh
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan.
J Theor Biol. 2005 Nov 7;237(1):104-16. doi: 10.1016/j.jtbi.2005.04.003.
Striped patterns are often observed on fish skin. Such patterns have been accounted for by reaction-diffusion (RD) Turing-type models, in which two substances can spontaneously form a spatially heterogeneous pattern in a homogeneous field. Among the striped patterns generated by Turing-type models, some are "straight-striped patterns," with many stripes running in parallel, while others are "labyrinthine patterns," in which the stripes often change direction, merge with each other, and frequently branch out. RD models differ in terms of their tendency to generate either labyrinthine or straight-striped patterns. Here, we studied the conditions under which either a labyrinthine or straight-striped pattern would emerge. First, we defined an index for stripe clearness, Sh. Straight-striped patterns (large Sh) are formed if only a narrow range of spatial periods corresponds to an unstable mode. Labyrinthine patterns (small Sh) are formed when a wide range of spatial periods is unstable. More specifically, labyrinthine patterns are formed when the maximum spatial period of unstable modes is more than twice that of the minimum spatial period of unstable modes; otherwise, straight-striped patterns are formed. We then examined RD models with nonlinear reaction terms, including both activator-inhibitor and substrate-depletion models, and we demonstrated that the same conclusions hold with respect to the conditions required for labyrinthine versus straight-striped patterns.
鱼类皮肤上经常能观察到条纹图案。这种图案可以用反应扩散（RD）图灵型模型来解释，在该模型中，两种物质能在均匀场中自发形成空间异质图案。在图灵型模型产生的条纹图案中，有些是“直条纹图案”，有许多平行的条纹，而另一些是“迷宫图案”，其中条纹经常改变方向、相互合并并频繁分支。RD模型在产生迷宫图案或直条纹图案的倾向方面存在差异。在这里，我们研究了迷宫图案或直条纹图案出现的条件。首先，我们定义了一个条纹清晰度指数Sh。如果只有狭窄范围的空间周期对应于不稳定模式，则形成直条纹图案（Sh值大）。当宽范围的空间周期不稳定时，形成迷宫图案（Sh值小）。更具体地说，当不稳定模式的最大空间周期大于不稳定模式最小空间周期的两倍时，形成迷宫图案；否则，形成直条纹图案。然后，我们研究了具有非线性反应项的RD模型，包括激活剂-抑制剂模型和底物消耗模型，并且我们证明了关于迷宫图案与直条纹图案所需条件的相同结论成立。
