Snails are model organisms for studying the genetic, molecular, and developmental bases of left–right asymmetry in Bilateria. However, the development of their typical helicospiral shell, present for the last 540 million years in environments as different as the abyss or our gardens, remains poorly understood. Conversely, ammonites typically have a bilaterally symmetric, planispiraly coiled shell, with only 1% of 3,000 genera displaying either a helicospiral or a meandering asymmetric shell. A comparative analysis suggests that the development of chiral shells in these mollusks is different and that, unlike snails, ammonites with asymmetric shells probably had a bilaterally symmetric body diagnostic of cephalopods. We propose a mathematical model for the growth of shells, taking into account the physical interaction during development between the soft mollusk body and its hard shell. Our model shows that a growth mismatch between the secreted shell tube and a bilaterally symmetric body in ammonites can generate mechanical forces that are balanced by a twist of the body, breaking shell symmetry. In gastropods, where a twist is intrinsic to the body, the same model predicts that helicospiral shells are the most likely shell forms. Our model explains a large diversity of forms and shows that, although molluscan shells are incrementally secreted at their opening, the path followed by the shell edge and the resulting form are partly governed by the mechanics of the body inside the shell, a perspective that explains many aspects of their development and evolution.
RESEARCH OF VIBRATION MACHINES DYNAMICS FOR PRODUCT SURFACES PROCESSING BY MATHEMATICAL MODELING
