Graph-Theoretical Analysis of Cleavage Pattern: Graph Developmental System and Its Application to Cleavage Pattern of Ascidian Egg: (graph theory/graph developmental system/cleavage pattern/ascidian egg).

The skeletal structure of the embryo is represented by the graph. In the graph, the cells and the connectivities between the cells are reduced to the nodes and edges, respectively. Along cleavage history, the series of graphs is obtained. In this paper I propose a new graph developmental system(GDS) which develops the series of graphs. And I represent and analyze the cleavage pattern of the ascidian egg by GDS. In order to represent it by GDS, at first, the connectivities between the cells are labeled according to the developments of the connectivities at the next time step, and next the cells are labeled. But there are two ways of labeling the cells, then two types of GDS are defined: (1) to label according to the pattern of the connectivities of their descendant cells after two time steps (G-GDS), (2) to label according to the cell fates (C-GDS). The C-GDS of the ascidian egg produces 16 cleavage patterns at 64-cell stage non-deterministically. If some labels are distinguished at 16-cell stage, the C-GDS becomes deterministic at 64-cell stage. GDS will be useful to simulate morphogenesis by the computer graphics.