Ellipse packing in 2D cell tessellation: A theoretical explanation for Lewis’s law and Aboav-Weaire’s law
- Published
- Accepted
- Subject Areas
- Cell Biology, Mathematical Biology
- Keywords
- Ellipse packing, Lewis's law, Aboav-Weaire’s law, 2D structures, tessellation, ellipse’s maximal inscribed polygon
- Copyright
- © 2018 Xu
- Licence
- This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction and adaptation in any medium and for any purpose provided that it is properly attributed. For attribution, the original author(s), title, publication source (PeerJ Preprints) and either DOI or URL of the article must be cited.
- Cite this article
- 2018. Ellipse packing in 2D cell tessellation: A theoretical explanation for Lewis’s law and Aboav-Weaire’s law. PeerJ Preprints 6:e27421v1 https://doi.org/10.7287/peerj.preprints.27421v1
Abstract
Background: To date, the theoretical bases of Lewis’s law and Aboav-Weaire’s law are still unclear.
Methods: Software R with package Conicfit was used to fit ellipses based on geometric parameters of polygonal cells of red alga Pyropia haitanensis.
Results: The average form deviation of vertexes from the fitted ellipse was 0±3.1 % (8,291 vertices in 1375 cells were examined). Area of the polygonal cell was 0.9±0.1 times of area of the ellipse’s maximal inscribed polygon (EMIP). These results indicated that the polygonal cells can be considered as ellipse’s inscribed polygons (EIPs) and tended to form EMIPs. This phenomenon was named as ellipse packing. Then, an improved relation of Lewis’s law for a n-edged cell was derived
\[cell\ area=0.5nab\sin(\frac{2\pi}{n})(1-\frac{3}{n^2})\]
where, a and b are the semi-major axis and the semi-minor axis of fitted ellipse, respectively. This study also improved the relation of Aboav-Weaire’s law
\[number\ of\ neighboring\ cells=6+\frac{6-n}{n}(\frac{a}{b}+\frac{3}{n^2})\]
Conclusions: Ellipse packing is a short-range order which places restrictions on the direction of cell division and the turning angles of cell edges. The ellipse packing requires allometric growth of cell edges. Lewis’s law describes the effect of deformation from EMIP to EIP on area. Aboav-Weaire’s law mainly reflects the effect of deformation from circle to ellipse on number of neighboring cells, and the deformation from EMIP to EIP has only a minor effect. The results of this study could help to simulate the dynamics of cell topology during growth.
Author Comment
This is a preprint submission to PeerJ Preprints.