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

Abstract

We developed a measurement framework of spatial organization to categorize 2-dimensional patterns from 2 multiscalar biological architectures. We propose that underlying shapes of biological entities can be approached using the statistical concept of degrees of freedom, defining it through expansion of area variability in a pattern. To help scope this suggestion, we developed a mathematical argument recognizing the deep foundations of area variability in a polygonal pattern (spatial heterogeneity). This measure uses a parameter called eutacticity. Our measuring platform of spatial heterogeneity can assign particular ranges of distribution of spatial areas for 2 biological architectures: ecological patterns of Namibia fairy circles and epithelial sheets. The spatial organizations of our 2 analyzed biological architectures are demarcated by being in a particular position among spatial order and disorder. We suggest that this theoretical platform can give us some insights about the nature of shapes in biological systems to understand organizational constraints.

Determining Levels of Biological Geometric Information at Polygonal Shape Patterns

“…Currently, it is widely accepted that although variation in those organizations exist there is just a narrow range of variations of cellular polygonal distributions [34], [40]. In this regard, samples of polygonal mesh are directly comparable even if these meshes are from different nature or scales [49], [50] due to all of them are PSP, including biological natural images, biological simulations, non-biological simulations, such as random meshes and Poisson-Voronoi tessellations and random polygons. Therefore, levels of Shannon entropy in polygonal meshes and set of random polygons turns out into a window of universal and comparable information if we approach them from a pure geometric perspective.…”

“…At tissue level, we used images from proliferating Drosophila prepupal wing discs (dWP); [34], [45], [47], middle third instar wing discs (dWL); [45], [47], normal human biceps (BCA); [34], muscular dystrophy from skeletal muscles (MD); [46] and pseudo stratified Drosophila wing disk epithelium (PSD); [40]. Also, at the ecological level polygonal meshes derived from Namibia fairy circles (ecological patterns associated with SDCP convergences [49]) images were integrated into the analysis (NFC); [49], [50], [51]. The global tag to define MD, dWP, dWL, BCA, PSD and NFC is called BIO.…”

“…We consider CS, SOE, SAE, AS and PT altogether as non-biological meshes (Non BIO), since they were derived from algorithms. Finally, we include an algorithmic routine from reference [49] to develop random arrangements [RA] into the global analysis as a control. Therefore, we have three mesh categories, BIO, Non BIO and RA.…”

Spatial Organization of Five-Fold Morphology as a Source of Geometrical Constraint in Biology

“…In fact, an important constraint in any spatial region may be the spatial homogeneity which we can understand as regularity and a statistical spatial organization argument of this constraint associated with FO is developed here. In a previous work, we considered quantitative spatial homogeneity among areas inside a region (a bounded polygon) as synonymous of regularity and the presence of disparity among areas as spatial heterogeneity [ 18 ]. Therefore, our definition of spatial heterogeneity is based on the unequal distribution of areas inside polygons; we propose a parameter to define quantitatively the spatial organization of inner polygonal elements around three main concepts: eutacticity, regularity and spatial heterogeneity.…”

“…Therefore, our definition of spatial heterogeneity is based on the unequal distribution of areas inside polygons; we propose a parameter to define quantitatively the spatial organization of inner polygonal elements around three main concepts: eutacticity, regularity and spatial heterogeneity. It has been shown in a previous work [ 19 ] that eutacticity is a parameter closely linked with regularity and it is a suitable measurement of spatial homogeneity and heterogeneity [ 18 ]. Eutacticity is sharply linked with regularity by considering that a given polygon, polyhedron and, in general, polytope can be associated with a star of vectors (pointing from the center to the vertices) and it has been demonstrated that stars associated with regular polytopes are eutactic [ 20 ].…”