It is often effective to build statistical models and to analyze the various random phenomena which occur in space from the point of view of space packing and tessellation. Among spatial events observed in nature, the randomly packed structure of many particles or the spatially tessellated structure of many cells appear at various fields. In particular, animal and human tissues which constitute the subjects of biology and medicine are full of the structures of packing and tessellation by cells. The main purposes of this research project are the statistical modeling and analyzing such structures through the mathematical elucidation, and at the same time, their heuristic adaptation to the actually observed spatial structures.
In this summary, we introduce our recent research about the rearrangement process of multicellular aggregate.

[vertex dynamics cell model]
The investigations of the deformation and the rearrangement of cell aggregates under the external force are important in the regeneration medicine and so on. If we assume the cells to be the polyhedra which divide the space without gaps, the geometrical structures of the cell aggregate are all represented by the information of the connection network among vertices and edges. Therefore, the spatio-temporal behavior of the cell aggregate can be given by the vertex dynamics. We are then able to give the equations of motion of vertices that rearrange the cells to minimize the total free energy.

[elementary process of reconnection of vertices]
For the topological change of the cellular polyhedra in accord with the motion of vertices, the reconnection of neighboring vertices should occur when a certain edge or a face becomes small (type H and type I in Fig. (a)). At that time, the corresponding faces change as in Figs. (b) and (c): a small face appears when the change from type I to type H happens and vice versa (Fig. (d)). This elementary process should be taken into account in the actual treatment of the equations of motion.
[computer simulations]
By solving the equations of motion of the vertex cell dynamics model under various conditions through Runge-Kutta method, many results are obtained which can explain the experimental evidences such as the rearrangement of cell aggregates from a flattened to a spherical structure due to the centrifugal force.

Members
Hisao Honda (Hyogo Univ.) and Tatsuzo Nagai
(Kyushu Kyoritsu Univ.).