Characterizing polygonality in biological structures.

Several systems involve spatial arrangements of elements such as molecules or cells, the characterization of which bears important implications to biological and physical investigations. Traditional approaches to quantify spatial order and regularity have relied on nearest neighbor distances or the number of sides of cells. The current work shows that enhanced performance can be achieved by considering angular regularity. Voronoi tessellations are obtained for each basic element and the angular regularity is then estimated from the differences between the angles defined by adjacent cells and a reference angle. In case this angle is 60 degrees, the measurement quantifies the hexagonality of the system. Other reference angles can be considered in order to quantify other types of spatial symmetries. The performance of the angular regularity is compared with other measurements including the conformity ratio (based on nearest neighbor distances) and the number of sides of the cells, indicating its improved sensitivity and discrimination power. The performance evaluation included synthetic (progressively perturbed hexagonal lattices) and real data (retinal mosaics). The good performance of the hexagonality measurements are illustrated also with respect to the problem of quantifying local spatial order in structures involving regions with different organizations as well as systems of points characterized by gradients of local order.