Pattern formation by two-layer Turing system with complementary synthesis.

Many multicellular organisms have a layered structure. The interaction between these layers plays an essential role in many developmental processes, and key molecules involved in these processes are often expressed in a layer-specific manner. On the other hand, pattern formation of organisms has been frequently discussed in connection with the Turing system. However, the Turing system has so far been studied mainly in single-layered space. In this paper, we thus investigate a two-layer Turing system with complementary synthesis, in which two interacting molecules are exclusively synthesized in different layers. From a linear stability analysis, we determine the Turing condition of the complementary system, and show that this condition requires stronger regulatory interactions of the molecules than that of the system with usual ubiquitous synthesis. We then confirm that this complementary system affects pattern types in fixed and expanding two-dimensional spaces in a similar way to the system with ubiquitous synthesis. In addition, the two-layer system includes two types of diffusion, lateral and transversal, and these have distinct effects on pattern formation with lateral diffusion mainly determining the periodicity of patterns generated and transversal diffusion affecting pattern type. These results suggest that the transversal diffusion functions as a time delay in the two-layer system. Finally, we apply this complementary system to explain pattern formation of the shoot apical meristem of plants. These findings provide an understanding of pattern formation caused by the interaction between cell layers in multicellular organisms.