Fig. 3.

Fig. 3.

Illustration of our topological techniques applied to zebrafish patterns. (A) Boundary conditions are periodic in the horizontal direction, so stripes and interstripes are viewed as loops from a topological perspective. (B) We count interstripes and measure stripe width using persistent homology. We show manifold expansions of the locations of Xd cells by considering balls of growing radius r centered at the location Xid of each cell. When r=r1, the radius of the balls is about half the maximum distance between neighboring Xd cells Δxx. At this point, three interstripes have formed, but the number of loops is larger than the true number of interstripes due to gaps between cells, highlighted by red arrows (β0=3 and β1>3). As r increases to r2, the noisy loops die off, leaving only three loops (β0=3 and β1=3). The long persistence of three loops corresponds to the true presence of three interstripes. As r increases further to r4, the manifold collapses to a single connected component (β0=1 and β1=1). The difference between the ball radius at which this collapse occurs (r4) and the ball radius at which three loops appear (r1) approximates half the maximum width of black stripes. (C) By combining TDA with clustering methods, we automatically detect interstripe boundaries and measure their curviness; we show the percentage of increase in arc length distance (ALD) of these boundaries (traced out in red) relative to perfectly straight stripes here. (D) We describe spotted phenotypes by combining persistent homology, clustering methods, and principal component analysis. We use β0 to quantify the number of spots. As an example, we show the spot size and spot roundness for two nacre spots.