Cells tile a flat plane by controlling geometries during morphogenesis of Pyropia thalli

Abstract

BackgroundPyropia haitanensis thalli, which are made of a single layer of polygonal cells, are a perfect model for studying the morphogenesis of multi-celled organisms because their cell proliferation process is an excellent example of the manner in which cells control their geometry to create a two-dimensional plane.MethodsCellular geometries of thalli at different stages of growth revealed by light microscope analysis.ResultsThis study showed the cell division transect the middle of the selected paired-sides to divide the cell into two equal portions, thus resulting in cell sides ≥4 and keeping the average number of cell sides at approximately six even as the thallus continued to grow, such that more than 90% of the cells in thalli longer than 0.08 cm had 5–7 sides. However, cell division could not fully explain the distributions of intracellular angles. Results showed that cell-division-associated fast reorientation of cell sides and cell divisions together caused 60% of the inner angles of cells from longer thalli to range from 100–140°. These results indicate that cells prefer to form regular polygons.ConclusionsThis study suggests that appropriate cell-packing geometries maintained by cell division and reorientation of cell walls can keep the cells bordering each other closely, without gaps.

Peer Review #2 of "Ellipse packing in two-dimensional cell tessellation: a theoretical explanation for Lewis’s law and Aboav-Weaire’s law (v0.1)"

“…structures. We found the average number of edges of P. haitanensis cells to be 6.00.9 (1375 cells in 13 thalli were examined; Table 1), which was consistent with previous studies on P. haitanensis as well as studies on many other organisms and physical structures (Gibson et al 2006;Sánchez-Gutiérrez et al 2016;Weaire & Rivier 1984;Xu et al 2017). According to Euler's 2D formula, this kind of phenomenon has been mathematically determined when the coordination number of each vertex equals three when different-size cells tessellate a 2D plane (Graustein 1931;Weaire & Rivier 1984).…”

“…Thallus of red alga P. haitanensis is a single-layered prismatic cell sheet that is a mathematical consequence of 2D expansion on a plane by cell proliferation (Xu et al 2017). Thus, P. haitanensis thalli can be simplified as 2D…”

Peer Review #3 of "Coccolith arrangement follows Eulerian mathematics in the coccolithophore Emiliania huxleyi (v0.1)"

“…Numerous studies have reported that the cell topology of many organisms follows mathematical rules. The two-dimensional (2D) Euler's formula was used in previous studies to explain why the average number of cell sides is six in many tissues, such as plant coverings, animal epithelia and seaweed (Gibson et al 2006;Lewis 1926;Xu et al 2017). The threedimensional (3D) Euler's formula was used to explain why the average face number of cells is approximately 14 in soap froth and many multicelled organisms (Lewis 1943;Weaire & Rivier 1984).…”

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“…To obey ellipse packing, the topological variations of Type I 2D structure need to be achieved by global adjustment (Büchner & Heyde 2017;Xu 2019); and that for Type II 2D structure can be achieved by local fine-turning, e.g. the division related allometric growth of cell edges of biological 2D structure (Xu 2019;Xu et al 2017). The cell growth kinetics of 2D structure also gained numerous scientific attentions.…”

“…However, in the above study, the basic geometric data of ten kinds of 2D structures, such as coordinates of vertices, edge number, and cell area, were derived from the images (Xu 2019). This kind of data collection may affect the analysis, for example, it is very difficult to separate points and very short edges (Xu et al 2017). To improve the analysis, we simulated a series of Voronoi diagrams by randomly disordering the seed locations of a regular hexagonal 2D structure following two previous studies (Zheng et al 2005;Zhu et al 2001).…”