[MS12] Mathematical and Computational Perspectives of Phyllotaxis and Growth of Plants

[MS12-04] Geometric Analysis of Differential Growth Governing 3D Morphogenesis of Biological Surfaces

*Kentaro Morikawa¹, Shinichi Morita^2,3, Kazuki Sakura^2,4, Akiteru Maeno⁵, Hiroki Gotoh⁶, Teruyuki Niimi^2,3, Yasuhiro Inoue¹ (1. Kyoto University (Japan), 2. National Institute for Basic Biology (Japan), 3. Graduate Institute for Advanced Studies, The Graduate University for Advanced Studies, SOKENDAI (Japan), 4. Japan Society for The Promotion of Science Research Fellowship (Japan), 5. National Institute of Genetics (Japan), 6. Shizuoka University (Japan))

Keywords:

Morphogenesis,Differential growth,Conformal map,Surface,Computational geometry

The development of complex three-dimensional (3D) structures in biological organs, such as plant leaves, flowers, and insect exoskeletons, occurs through surface growth. In the early stages of formation, tissues often have relatively simple, nearly flat shapes, which subsequently transform into curved surfaces. One of the key mechanisms driving this transformation is differential growth—spatially varying growth rates that lead to characteristic 3D morphologies. While the qualitative role of differential growth in shaping biological forms is well recognized, methods for quantitatively inferring growth rate distributions from observed shape changes have remained limited.In this study, we introduce a geometric method for inferring the differential growth that drives morphogenesis, specifically the spatial distribution of area expansion rates, from pre- and post-growth shapes. Our approach constructs a conformal map between the initial and final surfaces and computes the local area change at each point. This enables the inference of differential growth under the assumption of locally isotropic expansion. We applied this method to the developmental process of the horn primordium of the Japanese rhinoceros beetle Trypoxylus dichotomus. Our analysis revealed significant area expansion at the left and right edges of the horn primordium, consistent with experimental observations indicating a higher frequency of cell proliferation in these regions.These findings demonstrate the effectiveness of our approach in analyzing biological morphogenesis. Beyond its application to insect development, this method provides a quantitative geometric framework for studying plant leaf growth, facilitating a deeper integration of mathematical modeling and experimental observations.