Unveiling the role of differential growth in 3D morphogenesis: An inference method to analyze area expansion rate distribution in biological systems.

The three-dimensional (3D) morphologies of many organs in organisms, such as the curved shapes of leaves and flowers, the branching structure of lungs, and the exoskeletal shape of insects, are formed through surface growth. Although differential growth, a mode of surface growth, has been qualitatively identified as 3D morphogenesis, a quantitative understanding of the mechanical contribution of differential growth is lacking. To address this, we developed a quantitative inference method to analyze the distribution of the area expansion rate, which governs the growth of surfaces into 3D morphology. To validate the accuracy of our method, we tested it on a basic 3D morphology that allowed for the theoretical derivation of the area expansion rate distribution, and then assessed the difference between the predicted outcome and the theoretical solution. We also applied this method to complex 3D shapes and evaluated its accuracy through numerical experiments. The findings of the study revealed a linear decrease in error on a log-log scale with an increase in the number of meshes in both evaluations. This affirmed the reliability of the predictions for meshes that are sufficiently refined. Moreover, we employed our methodology to analyze the developmental process of the Japanese rhinoceros beetle Trypoxylus dichotomus, which is characterized by differential growth regulating 3D morphogenesis. 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. Hence, these findings confirm the efficacy of the proposed method in analyzing biological systems.