  Most of normal proliferative epithelia of plants and metazoans are
topologically invariant and characterized by similar cell distributions
according to the number of cell neighbors (DCNs). Here we study peculiarities
of these distributions and explain why the DCN obtained from the location of
intercellular boundaries and that based on the Voronoi tessellation with nodes
located on cell nuclei may differ from each other. As we demonstrate, special
microdomains where four or more intercellular boundaries converge are
topologically charged. Using this fact, we deduce a new equation describing the
topological balance of the DCNs. The developed theory is applied for a series
of microphotographs of non-tumoral epithelial cells of the human cervix
(HCerEpiC) to improve the image processing near the edges of microphotographs
and reveal the topological invariance of the examined monolayers. Special
contact microdomains may be present in epithelia of various natures, however,
considering the well-known vertex model of epithelium, we show that such
contacts are absent in the usual solid-like state of the model and appear only
in the liquid-like cancer state. Also, we discuss a possible biological role of
special contacts in context of proliferative epithelium dynamics and tissue
morphogenesis.
