Local growth laws determine global shape of molluscan shells
- 1Division of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125
Abstract
Molluscan shells come in various shapes and sizes. Despite this diversity, each species produces a shell with a characteristic shape that is independent of environmental conditions. We seek to understand this robust complexity. We are guided by two principles in the spirit of D’Arcy Thompson. First, the growth is governed by the repeated and continuous application of a fixed growth law, even as the shell evolves in overall shape, without any complex biological machinery to monitor and control the growth. Second, the growth law depends solely on local geometry at the shell’s growing edge. The first principle naturally leads to the mathematical statement that the shape of the shell is generated by the action of a Lie group on a protoconch. The second naturally leads to a particular representation of the Lie group. We use this representation to show that the shapes of nearly all known molluscan shells can be described by essentially three parameters: a scalar (scaling), a vector (orientation), and a curve (edge of the protoconch). We relate these parameters to the phylogenetic tree. In addition to the morphogenetic insight, our results potentially point to a alternative growth-based approach to engineering complex structures.
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BibTeX
@article{liu2026local,
title = {{Local growth laws determine global shape of molluscan shells}},
author = {Huan Liu and Kaushik Bhattacharya},
journal = {Proceedings of the National Academy of Sciences of the United States of America},
year = {2026},
month = oct,
doi = {10.1073/pnas.2614478123},
url = {https://doi.org/10.1073/pnas.2614478123},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - JOUR
TI - Local growth laws determine global shape of molluscan shells
AU - Huan Liu
AU - Kaushik Bhattacharya
PY - 2026
DA - 2026/10/23
JO - Proceedings of the National Academy of Sciences of the United States of America
DO - 10.1073/pnas.2614478123
UR - https://doi.org/10.1073/pnas.2614478123
AB - Molluscan shells come in various shapes and sizes. Despite this diversity, each species produces a shell with a characteristic shape that is independent of environmental conditions. We seek to understand this robust complexity. We are guided by two principles in the spirit of D’Arcy Thompson. First, the growth is governed by the repeated and continuous application of a fixed growth law, even as the shell evolves in overall shape, without any complex biological machinery to monitor and control the growth. Second, the growth law depends solely on local geometry at the shell’s growing edge. The first principle naturally leads to the mathematical statement that the shape of the shell is generated by the action of a Lie group on a protoconch. The second naturally leads to a particular representation of the Lie group. We use this representation to show that the shapes of nearly all known molluscan shells can be described by essentially three parameters: a scalar (scaling), a vector (orientation), and a curve (edge of the protoconch). We relate these parameters to the phylogenetic tree. In addition to the morphogenetic insight, our results potentially point to a alternative growth-based approach to engineering complex structures.
SN - 0027-8424
SN - 1091-6490
ER -