Boundary effects in biological planar networks: pentagons dominate <i>Pyropia</i> marginal cells

The topological and geometrical features at the boundary zone of planar polygonal networks remain poorly understood. Based on observations and mathematical proofs, we propose that marginal cells in the thalli of Pyropia haitanensis, a two-dimensional (2D) biological polygonal network, have an average edge number of approximately five. We demonstrate that this number is maintained by specific division patterns. Furthermore, we observe that both marginal cells and inner cells follow the trends predicted by the Lewis law and Aboav-Weaire law, but each cell type requires its own set of correlation parameters to more accurately describe its topological and geometrical features. The boundary effects are also evident in the differences between marginal cells and inner cells in terms of the distributions of interior angles and edge lengths. Similar to inner cells, cell division tends to occur in marginal cells with large sizes and transects a pair of unconnected edges. In particular, this study finds that the division of marginal cells preferentially transects the marginal edge. These specific topological and geometrical features of marginal cells and division patterns may inform the development of modelling algorithms for boundary conditions in biological 2D cellular networks.