Finite element analysis of a double membrane tube (DMS-tube) and its implication for gastropod shell morphology

Molluscan shells, including those of Gastropoda, are formed by accretionary growth at the mantle edge. The mantle is a thin membrane of skirt-like shape, which extends minutely beyond the aperture, and its edge adds a shell increment to the aperture margin so that each increment copies a configuration of the mantle edge at that time. Thus, regulation of shell morphogeny is almost equivalent to the factors which control the mantle form at the moment of shell growth. Form of the mantle skirt is considered to be kept in a state of balance between the force of its internal stress and forces acting on it such as fluid pressure or muscle contraction. The expansion behavior of the mantle skirt has been numerically analyzed by using an elastic model (DMS-tube), which represents the fundamental structure of the mantle tissue as a double membrane structure with internal springs (DMS). Four characteristic expansion patterns of the DMS-tube have been detected: (1) general outward expansion; (2) developing a ridge-like fold on an initial longitudinal protrusion of the tube edge; (3) drastic shift of the expanded state from a uniformly curved to an elliptical shape in outline, owing to the existence of a fixed boundary condition on the tube wall; and (4) constricted protrusion on the open region of the shell wall surrounding the DMS-tube. These results have the potential for answering the following questions relating to the morphogenesis of gastropod shells. How does the mantle skirt usually make contact with the inner surface of the shell wall so as to ensure continuous accretion of shell materials to the aperture margin? What is the cause of spiral ridges? Why do open coiling or minimally overlapping shells have generally circular apertures, while shells with apertures overlapped by whorls have non-uniformly curved apertural lips? What is the cause of long closed spines and why do they always appear on spiral ridges?