Characterizing symmetry transitions in systems with dynamic morphology.

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.