Geometric analysis and estimation of the growth rate gradient on gastropod shells

The morphology of gastropod shells provides a record of the growth rate at the aperture of the shell, and molecular biological studies have shown that the growth rate gradient along the aperture of a gastropod shell can be closely related to gene expression at the aperture. Here, we develop a novel method for deriving microscopic growth rates from the macroscopic shapes of gastropod shells. The growth vector map of a shell provides information on the growth rate gradient as a vector field along the aperture, over the growth history. However, it is difficult to estimate the growth vector map directly from the macroscopic shape of a specimen, because the degree of freedom of the growth vector map is very high. In order to overcome this difficulty, we develop a method of estimating the growth vector map based on a growing tube model, where the latter includes fewer parameters to be estimated. In addition, we calculate an aperture map specifying the magnitude of the growth vector at each location, which can be compared with the expression levels of several genes or proteins that are important in morphogenesis. Finally, we show a concrete example of how macroscopic shell shapes evolve in a morphospace when microscopic growth rate gradient changes.