A Method for Inferring Differential Growth Rate Distributions in 3D Surface Morphogenesis Using Conformal Maps

Biological organs such as flowers, lungs, and insect exoskeletons develop intricate three-dimensional (3D) structures through surface growth. A key mechanism underlying the transformation from initially flat or simple shapes into complex 3D forms is differential growth, characterized by spatially varying growth rates. While the qualitative role of differential growth in morphogenesis is recognized, a quantitative understanding of how these growth rate distributions drive 3D shape formation is still lacking. In this study, we present a novel method to quantitatively infer the distribution of growth rates responsible for 3D morphogenesis. Our approach constructs a conformal map between pre- and post-growth shapes, allowing us to extract the spatial growth rate distribution. We applied this method to the developmental process of the horn primordium in the Japanese rhinoceros beetle Trypoxylus dichotomus. The results indicated a notably high rate of area expansion on the left and right edges of the horn primordium, which is consistent with the experimental evidence of a higher rate of cell division in these regions. This result demonstrates the efficacy of our method in quantitatively analyzing differential growth in biological systems, offering new insights into the mechanics of morphogenesis.