Topology of dividing planar tilings: Mitosis and order in epithelial tissues

We investigate a range of rule-based models of the in-plane structure of growing single-cell-thick epithelia represented by the distribution of frequencies of polygon classes. Within the Markovian framework introduced by Gibson et al. [Nature (London) 442, 1038 (2006)], we discuss various topologically allowed cell division schemes assumed to control the structure of the tissue as well as a phenomenological Gaussian scheme, and we compute the stationary distributions for all of them. Some of the distributions reproduce those seen in tissues characterized by unbiased mitotic events but also in certain tissues with a preferred orientation of the mitotic plane or a cell-rearrangement process such as neighbor exchange. In addition, we propose the asynchronous-division variant of the model, which builds on the Lewis law and on the Aboav-Weaire law as well as on the fact that the dividing cells are larger than the resting cells. This generalization a posteriori validates the original model.