A measure of regularity for polygonal mosaics in biological systems

Abstract

BackgroundThe quantification of the spatial order of biological patterns or mosaics provides useful information as many properties are determined by the spatial distribution of their constituent elements. These are usually characterised by methods based on nearest neighbours distances, by the number of sides of cells, or by angles defined by the adjacent cells.MethodsA measure of regularity in polygonal mosaics of different kinds in biological systems is proposed. It is based on the condition of eutacticity, expressed in terms of eutactic stars, which is closely related to regularity of polytopes. Thus it constitutes a natural measure of regularity. The proposed measure is tested with numerical and real data. Numerically is tested with a hexagonal lattice that is distorted progressively and with a non-periodic regular tiling. With real data, the distribution of oak trees in forests from three locations in the State of Querétaro, Mexico, and the spiral pattern of florets in a flowering plant are characterised.ResultsThe proposed measure performs well and as expected while tested with a numerical experiment, as well as when applied to a known non-periodic tiling of the plane. Concerning real data, the measure is sensitive to the degree of perturbation observed in the distribution of oak trees and detects high regularity in a phyllotactic pattern studied.ConclusionsThe measure here proposed has a clear geometrical meaning, establishing what regularity means, and constitute an advantageous general purposes alternative to analyse spatial distributions, capable to indicate the degree of regularity of a mosaic or an array of points.

A Method to Categorize 2-Dimensional Patterns Using Statistics of Spatial Organization

“…Sample 1 (from epithelial topology lattice or set gamma 4; Figure 10B ) was developed using the centroids derived from polygonal coordinates of each cell which were the points for the Voronoi tessellation treatment that will constitute the seeds for Voronoi construction that yields a new set of polygons. Regularity in sample 1 (from set gamma 3) increased to 0.947; this agrees with the results of Contreras-Figueroa et al 17 As a conclusion, sample 0 and sample 1 have the same number of polygons, and their polygonal frequency is also very close to the UEF. Nevertheless, in terms of regularity, they are not the same; sample 0 is 0.929 and sample 1 is 0.947.…”

“…The breaking of spatial homogeneity or regularity in a lattice is called spatial heterogeneity, and we propose a statistical parameter to define quantitatively the spatial organization of 2D patterns. Associating this parameter with eutacticity (it has been shown in a previous work by Contreras-Figueroa et al 17 that eutacticity is closely linked with regularity and is a suitable measurement of spatial heterogeneity), we proved that there are exclusive biological pattern properties that are different from those found in nonbiological architectures. In fact, according to our proposed parameter, biological spatial organizations are characterized by being in a particular position among spatial order and disorder.…”

“…However, to provide a practical measure of regularity using the statistical framework described, we conduct a numerical proof relating MGD and GDG to a mathematical measure called eutacticity in section “Numerical proof associating eutacticity and standard deviation of dispersion mean of a module.” Eutacticity has been associated with regularity in terms of spatial distribution of areas. 7 Although eutacticity has been proposed as a measure of regularity and appears to be a meaningful property to understand spatial order, 17 it has not been properly explained as a source of information to characterize 2D patterns statistically, establishing particular limits for biological organizations. As a result of our numerical proof, we realize that the statistical framework given before provides an explanation of the meaning of the eutacticity parameter.…”

“…Regarding the epithelial tissues, several arguments have been put forward questioning the value of regularity associated with particular polygonality. [17][18][19][20] Sánchéz-Gutiérrez et al (2016) 18 describe how Voronoi diagrams are able to predict the diverse polygon distributions of any natural packed tissue. In that work, the authors demonstrate that Voronoi tessellations and the very different tissues analyzed share a restriction: the frequency of polygonal types correlates with the distribution of cell areas.…”

“…Section “Mathematics of eutacticity” is about mathematics of eutacticity considering important previous references. 17,26 Our numerical proof in section “Numerical proof associating eutacticity and standard deviation of dispersion mean of a module” will define the amplitude of spatial variability of elements in a lattice, called modules, showing the association between eutacticity with our statistical argument. In section “Generation of global form normalΓ ,” the method to build global forms called sets gamma is shown, and the statistical categorization of them is given.…”