Computational topology of Drosophila melanogaster embryos: Emergent anatomy from the spatiotemporal distributions of hemocytes

Hemocytes migrate within embryos during their morphogenesis. The spatial distribution of the hemocytes reflects the emergent anatomy of the embryos. It poses a significant challenge to statistical characterization using conventional methods, such as mean-square displacements of tracks and Ripley's <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>L</a:mi></a:mrow></a:math> function. This challenge arises from the complex heterogeneity of hemocyte dynamics and the variability of the organism's anatomy due to inherent nonlinear stochastic processes. We apply topological data analysis (TDA) to three different genotypes of embryos: wild type, <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>L</b:mi><b:mi>a</b:mi><b:mi>n</b:mi><b:mi>B</b:mi><b:msup><b:mn>1</b:mn><b:mrow><b:mi>D</b:mi><b:mi>f</b:mi></b:mrow></b:msup></b:mrow></b:math> (extracellular matrix mutants), and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>S</c:mi><c:mi>C</c:mi><c:mi>A</c:mi><c:msup><c:mi>R</c:mi><c:mrow><c:mi mathvariant="normal">Δ</c:mi><c:mn>37</c:mn></c:mrow></c:msup></c:mrow></c:math> (motility mutants). Computationally efficient methods are developed to characterize the embryos' topologies. Based on the Wasserstein distances, the mutant is topologically distinct from the wild type and mutant, which are indistinguishable. Furthermore, an increase in the Betti-1 number (the number of holes in the hemocyte configurations) is observed across all samples with age, which is consistent with prior findings on cellular tiling behavior due to contact inhibition of locomotion. We also apply TDA methods to the hemocyte tracks within the embryos, allowing the wild type and mutant to be distinguished. Computational topology thus provides a robust method to quantify the emergent anatomy of embryos that is well suited to the complex heterogeneous datasets observed with hemocyte dynamics.