Approximate symmetry
References
2025
- Math. Biosci.Characterizing symmetry transitions in systems with dynamic morphologyMathematical Biosciences, 2025
The accurate quantification of symmetry is a key goal in biological inquiries because symmetry can affect biological performance and can reveal insights into development and evolutionary history. Recently, we proposed a versatile measure of symmetry, transformation information (TI), which provides an entropy-based measure of deviations from exact symmetry with respect to a parameterized family of transformations. Here we develop this measure further to quantify approximate symmetries and maximal symmetries represented by critical points in TI as a function of a transformation parameter. This framework allows us to characterize the evolution of symmetry by tracking qualitative changes with respect to these critical points. We apply TI to increasingly complex settings, from mathematically tractable probability distributions to differential equation models with emergent behaviors that are inspired by developmental biology and formulated in both static and growing domains. Our analysis of the qualitative changes in symmetry properties indicates a potential pathway toward a general mathematical framework for characterizing symmetry transitions akin to bifurcation theory for dynamical systems. The results reveal deep connections between observed symmetry transitions, subtle changes in morphology, and the underlying mechanisms that govern the dynamics of the system.
@article{ciocanel2025characterizing, title = {Characterizing symmetry transitions in systems with dynamic morphology}, journal = {Mathematical Biosciences}, volume = {384}, pages = {109431}, year = {2025}, issn = {0025-5564}, doi = {https://doi.org/10.1016/j.mbs.2025.109431}, url = {https://www.sciencedirect.com/science/article/pii/S0025556425000574}, author = {Ciocanel, M.-Veronica and Gandhi, Punit and Niklas, Karl and Dawes, Adriana T.}, keywords = {Approximate symmetry, Developmental biology, Information theory, Bifurcations, Pattern formation} }
2021
- Phil. Trans. AIdentification of approximate symmetries in biological developmentPhilosophical Transactions of the Royal Society A, 2021
Virtually all forms of life, from single-cell eukaryotes to complex, highly differentiated multicellular organisms, exhibit a property referred to as symmetry. However, precise measures of symmetry are often difficult to formulate and apply in a meaningful way to biological systems, where symmetries and asymmetries can be dynamic and transient, or be visually apparent but not reliably quantifiable using standard measures from mathematics and physics. Here, we present and illustrate a novel measure that draws on concepts from information theory to quantify the degree of symmetry, enabling the identification of approximate symmetries that may be present in a pattern or a biological image. We apply the measure to rotation, reflection and translation symmetries in patterns produced by a Turing model, as well as natural objects (algae, flowers and leaves). This method of symmetry quantification is unbiased and rigorous, and requires minimal manual processing compared to alternative measures. The proposed method is therefore a useful tool for comparison and identification of symmetries in biological systems, with potential future applications to symmetries that arise during development, as observed in vivo or as produced by mathematical models. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.
@article{gandhi2021approximate, author = {Gandhi, Punit and Ciocanel, M.-Veronica and Niklas, Karl and Dawes, Adriana T.}, title = {Identification of approximate symmetries in biological development}, journal = {Philosophical Transactions of the Royal Society A}, volume = {379}, number = {2213}, pages = {20200273}, year = {2021}, doi = {10.1098/rsta.2020.0273}, url = {https://royalsocietypublishing.org/doi/abs/10.1098/rsta.2020.0273}, eprint = {https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2020.0273} }